First of all, if you can avoid using Skype, do so, as it has been backdoored by Microsoft for the NSA and is no longer secure, and does not provide the best experience even if you want your convo to be public (see below).

Better than Skype

If you want your video chat to be public, use Google Hangouts instead, which gives you the option of saving the video to YouTube.

If you want your video chat to be private, use something encrypted like or Jitsi, which has a convenient web version.

Skype silently dies on Linux?

See if it's still running with

ps aux | grep skype

If it's already running, kill it with something like

sudo kill -9 $(ps aux | grep skype | awk '{print $2}')

Now it should work. If it doesn't, remove the lock files and try again; see this and this for help doing so.