I enjoyed both games but there's no real competition here, Skyrim wins hands down, more-so if you are playing it on a PC and can enjoy all them fancy mods it has.
Kingdoms of Amalur was fun but it was always designed as a pre-cursor to something bigger. This fact always left a really bitter taste in my mouth as KoA was fun but could've been so much more. The worst part is that patches that were developed to fix bugs and the like will likely never get released to the developer going under, which honestly just makes Skyrim that much more of the winner in this scenario.
Gameplay wise KoA is actually better than Skyrim. This is partially because as repetitive as it can get, Skyrim's combat becomes just as tedious and repetitive but at least KoA's tries to be slightly different whereas Skyrim's is still the same old similar routine from previous Elder Scrolls games.
Storyline-wise KoA wins over Skyrim. I thought Skyrim had plenty of potential but completely failed to live up to it. The idea of an actual Dragon War was brilliant but it never happened. Alduin was an absolute joke, only ever seeing him three times and in 2/3 he was a complete push-over. KoA's storyline wasn't the best but it was far more epic and it's plot twists far more exciting.
Overall Skyrim is a better game simply because you'll get far more for your money. Both games can grow repetitive and tedious grinds after awhile but Skyrim less so than Kingdoms of Amalur. Kingdoms of Amalur, in many cases, feels far too "MMO-y", because that's what it was a pre-cursor for, but that just lets it down.
