Green screen without green screen
Yes, that is possible. What you're searching for is a difference matte / difference key. For that technique, you supply a base image and the program keys everything out that equals that image. So you could key out your static background so that only you remain in the picture.

However, there are a few downsides:


This technique doesn't work nearly as good as a greenscreen. It will probably require a lot of work and there will still be some artifacts left in the video.
If you're using a bad camera (such as a laptop webcam), the result will be even worse.
If you move your camera even slightly, you'll have a problem. So if you do that with a webcam, don't touch it at all, if possible.
There probably aren't many applications that can do this effect live (as opposed to post-production). IIRC, there is some video chat app for Mac-Computers that can do this. I'm not sure if it is possible with Skype, you'll have to research that. However, keep in mind that it probably won't look as good as you imagine it.