. . . . Windows media player plays it without a problem . . . .
As @CubeAce has commented: Just because a video will play in a playback program will not mean it is editable.
The Skype video sync issue is probably due to one or both of the following:
recorded using a variable framerate (VFR) for the video, this gives rise to the timing issue of syncing the audio to the video for many video editors which prefer a Constant framerate (CFR) for the video.
the sample frequency is different from the video standard of 48kHz such that the audio may be 'stretched' or shortened depending on the sample rate difference from the expected
In both cases the video needs converting to CFR for the video and 48 kHz sample rate for the audio - this can be done in a video converter such as Handbrake.
Download and install MediaInfo and analyse one of the clips and post the results, see this tutorial on how to setup MediaInfo and analyse a video clip for all the data required and post this in a reply.
It would also help if you put your computer specification and program version in your profile signature to give us an overview of the hardware involved and, in future posts, so we do not have to keep asking for it, - see this topic for what is required and please quote processor and graphics card make/model in full.