Open Skype and get to the point where the "being used by another program" appears. Then use Task Manager to stop each running application as applicable one at a time. Watch for the message to stop. The goal is to identify what program is using the webcam or at least what program Skype believes is using the webcam.
Failing that:
First check the webcam's status via Device Manager. The webcam could have information/settings in two places: Imaging Devices and Sound, Video, Game Controllers.
As always, verify that you have the correct webcam drivers for your OS from the manufactuer for your model camera.
Also click the speaker icon probably right bottom corner of your screen. Go into Recording Devices and select your webcam. Then go into the properties settings. Check all of the tabs and settings. Check the Communications tab.
Take some time to explore and look at all of the various settings. Very easy to have some misconfiguration via poor end user documentation, not-so-specific configration windows, software resetting defaults, or just some user oversight or misunderstanding.
Change things only one at a time and keep notes so you can reverse a setting if necessary.