I want to live feed what I'm doing on Skyrim through Skype to send to someone.
Skype has no built in recording feature buy you can use a 3rd party program to record video calls. You can see some recording programs that are available here:
you can try Supertintin (which is also in the Skype Apps Directory) as you are using Windows. This recorder can also record Skype screensharing session.http://shop.skype.com/apps/Call-recording-audio-video/SuperTintin-Video-Call-Recorder.html