I would like to install Skyrim also on my desktop. I was wondering if Steam will let me install Skyrim on another PC also? If not, I don't want to go through setting it up and downloading it, all 7 or so GB.
Yes, you can install steam games on as many computers as you want.
The catch, is that you may only be connected to steam from one computer at a time. For a game like Skyrim, however, running in Steam's Offline mode isn't much of an issue.