Skype has a WebUI that does not allow one to use Firefox for some reason. Lucky for us, the check is done with the User Agent. So we can spoof that.

Spoof User Agent

1. First install the User-Agent Switcher and Manager Addon
2. Optional: Enable the addon in private Windows
3. Preferences: Switch to White-List Mode and set these domains skype.com, live.com, microsoft.com and press Save
4. Toolbar Icon: Select the latest Chrome from the list
5. Navigate to web.skype.com (in a private Window if you prefer)