With an updated Skype for Web service, Microsoft has eliminated support for all but the Edge and Chrome browsers on certain hardware. To get around this on a Chromebook, you can use this trick.
After making good progress on audio support for Linux apps on Chrome OS, the feature appears to have missed the cut for Chrome OS 73: Tune in to Chrome OS 74, at the earliest, for audio playback in Project Crostini.
Sound support is still in the works for Linux apps on Chromebooks, but the first function will simply be that: Sound playback. Audio recording is slated to follow and camera access — needed for Skype video — isn’t even on the roadmap yet.