Do you have any other app installed that's using the camera? I'm just brainstorming, but wondering if there's an interference there. I have Camera Zoom FX installed, but other than that no other camera app.
So it's not another application interfering and you've tried uninstalling and reinstalling. You have verified both cameras work I assume.
I guess you could try another Skype application. I believe there is another one. If that doesn't work and no one else has a suggestion, I'd try Skype support.