For better screenshots in Skyrim, you should first turn off the HUD. This removes the compass, the cross-hair, destination markers, and other screen objects. Just open the developer console by pressing tilde (~), then type in tm and hit enter. The HUD will disappear, including the developer console itself. The game will still be frozen, but don't worry it hasn't crashed. It's because the dev console is still up. Hit tilde again to leave the dev console and resume play.

Once you're done taking screenshots, you can bring the HUD and everything back by going through the same process. Though this time you'll have to do it blind. Hit tilde and the game should freeze, though you won't see the develop console. Type in tm just like before and hit enter. If you did it right, everything should pop back up. If it doesn't, hit tilde again to exit the dev console, and then try again.

For those wondering, you can take screenshots in-game by pressing the Print Screen button.