Awhile back I bought Skyrim for my PC, and the other day when I pressed "Play" in Steam, it sent me to the page to buy it, and when I tried going to the button to see what happens if I buy it again, it says I already own the game.
This was likely due to a temporary issue with Steam on August 4th. From the Steam Twitter account:
We've fixed an issue where the Steam client wasn't displaying ownership correctly for some games.
You may need to restart your Steam client for the changes to take effect. Apologies for the disruption!
There are multiple reports on that thread of this happening with Bethesda games, and speculation that it was occurring for games which were on sale, which would have included Skyrim.
You should be able to play now, though as the above message states, you may need to restart Steam.