well, I had the same problem and found a solution

)
- go to steam
- select skyrim (naturally--> what else?

)
-open the properties (click with right key)
-choose the register local files
- click third option (check gamefiles for errors or something like that (sorry mine is in german

))
if you're lucky there are 2 files which could not be checked and they will be downloaded anew --> tadaaaah you can play

hope everything works

)