I'd pick DAI over Skyrim, but not by that much (Fallout to me is a much better game than either of these two). But to be fair, I'll just list out all my likes and donts for each game.
DAI Likes:
Engaging story - OMG I was so intrigued by that giant big hole that I just wanted to go inside it and see what was waiting for me on the other side (yes I know what that sounds like). Not to mention the wide expanse of lovely characters that you meet along the way
Enjoyable characters (excluding Sera considering she's such a disrespectful snob) - The characters just engage me much more than Skyrim's "Hello fellow Companion. I would like you to give this shield to that guy on the forge that is literally only 10 steps away". They have such a diverse cast, from the tough-hearted Blackwall to the sweetheart Josephine.
Much better combat - The combat in DAI just feels so much smoother, the way that the animations all come together, along with the ability to use multiple abilities at once
Dragon fights - The dragon fights feel so organic, with each dragon moving around the battlefield, using different tactics, such as clawing at you, or flying away to fire their elemental powers from afar. They just don't sit their and... bite at you.
Variety of the enemies - From Red Templars to simple bandits to Pride Demons to giants, this game has a very unique choice of enemies for you to fight. Skyrim? One word: Draugr.
Stunning world - DAI has such a beautiful world, from the luscious forests of the Emerald Graves, to the Red Lyrium ridden snowy planes of Emprise du Lion, DAI really provides eye candy when it comes to a large world.
DAI DONTS:
Tactical cam - Really BW? You couldn't make the tac cam work? I mean it was fine in DAO and DA2, but if I try to use it on DAI, it just kinda flies around the place while trying to show you the penny on the ground.
Weapon and armor diversity - In each of my playthroughs, I'm always stuck with the same weapon and armor for 60% of the game. One playthrough, I got Tempest pretty early in the game, and I didn't even change it, even post Corypheus fight.
Disappointing side quests - Almost every side quest I've experienced was either a fetch quest or a quest where you go and deliver something. But there are also some side quests I actually enjoyed, for example that one in the HW where you had to find the little parts of Fairiel's Key, which was nice because it told a pretty good story while involving a unique puzzle into the mix.
Skyrim Likes:
Weapon and armor diversity - The amount of different unique weapons you can find during Skyrim and its expansion packs is HUGE. I mean, you start with some pretty dull armor but you eventually end up with glass armor and the eternally elegant ebony armor.
Cities - The cities in Skyrim felt large and alive (they're pretty small compared to Oblivion's Skingraad and Imperial City, but this is about Skyrim, not Oblivion), not to mention the different personalities in them, from beggars to stuck-up royalties
Detail to the world - The world of Skyrim has so much detail in every single cranny that it's a wonder how Bethesda even got it out within 5 years away from its predecessor, Oblivion. Each dungeons has its own little backstory, each book you read with its own lore and characters.
Skyrim DONTS:
Guild quests - To sum it up, "Oh hey, wassup newbie! Yea, our old leader kinda died while you were gone, but you know what, since you got us those pieces for this axe that we wanted REALLY bad, why don't you come here and become our new leader. Oh, why can't I be the leader even though I've been here ever since I was a kid? We'll give you the leadership since you're so trustworthy after those few days we've known you even though you might jeopardize our lives during your tangle with the entire Imperial forces."
Boring combat - Each time I go to load Skyrim, I'm always so excited to keep spamming the attack button! NO. The combat in Skyrim is so repetitive and boring it always turns me off after a few hours. All the combat is is just you spamming attack until the person you're locked into is dead. No change, just keep swinging that sword. The fact that it's nearly always the same animation each time makes me die in the inside a bit.
Dungeons - Skyrim really has some nice quests (except for guild quests), but the fact each dungeon has draugrs in it is a big let down for me. Why couldn't Bethesda put in a more diverse cast of enemies. Throw in some dremora in their, maybe a few goblins or wraiths. But noooo. It has to be draugr EVERY. SINGLE. TIME.
Dragon fights - The first time I saw Mirmulnir, I was so hyped to kill my first dragon! But after of while of tediously hacking away, it just kinda died. I was like "that's it? It didn't even do any cool moves or unique movesets! It just flew around and breather stuff at me!" The biggest disappointment other than the final fight.
Each game has its ups and downs, but for me DAI takes the cake always. With the way it presents its world to us, it is always such a refresher to my mind that I don't have to fight any more draugr, and just get to sit back and relax with the enjoyable banter between the likes of Cassandra and Varric. Don't forget that the combat always keeps me glued to the screen with its flashy animations and wide range of sound effects (though they could have worked on tac cam, as it played such a big role in DAO, personally my favorite out of the trilogy). Skyrim is still a good game, but they could have improved with the combat and the personalities of your companions, not to mention those horrid dragon fights.