First try all relevant suggestions from Apple's http://support.apple.com/kb/HT2090
Test using more than one Apple app in more than one user account.
If your iSight works with the Apple apps in every user account you test, check Skype again to verify whether your problem still exists with Skype. If it does, be sure that both your Apple software and Skype are up-to-date.
For Skype-specific problems that remain even after updating to the latest Skype version that is compatible with your Mac OS, Skype's comprehensive support system should best be able to help you use their product with your system.
Start here: https://support.skype.com/
You can also search or post for help with your problem in Skype's dedicated Skype Discussions for Mac. The Skype users who post there may have already solved the same problem.
If nothing else works, you can even sign in and ask for email help directly from Skype's Technical Support people.