Camera will not work in Skype in 10.8.5
Apple issued a patch to go on top of the 10.8.5 update that didn't ship both a 32bit and 64 bit file that Skype, Apple's iMovie and other apps need to run the internal camera.

After some investigation, we have found that the OSX 10.8.5 update is missing a specific file that would allow some apps such as Skype or iMovie to access the built-in facetime camera on Mid-2013 MacBook Airs.
The 10.8.5 update updated a CoreMediaIO plugin that accesses the MBA camera, but didn't bundle the 32-bits version of it.

Source: Skype Community Post
Also, you can use an external camera in the mean time if you need to run on 10.8.5 and the affected Air.