It turns out that the Windows 8.1 breaks down Skype installation
In addition to many other problems experienced after deploying Windows 8.1, it turns out that the new OS update released by Microsoft last week is also causing trouble to Skype users, as the Modern version of the app stops working unexpectedly.Users are now posting on Microsoft’s support forums to reveal that Skype no longer works after making the switch to Windows 8.1 due to reasons that are yet to be determined.
“I upgraded my system to Windows 8.1 from Windows 8 yesterday. Since then I could not log on to Skype. Few seconds after Skype tries to connect, the system returns to the Start screen with other tiles,” one of the affected users posted.
It turns out that the Skype team is aware of these issues, but no workaround is available at this time, so your only option is to remove and re-install the Skype app on Windows 8.