Finding out your Skype ID is easy and you have 2 options to do so:
1. You go to the main page where you can register a new account:
https://accounts.skyprivate.com/users/register/ and click on "Don't know your Skype ID? Find it here."
You fill in your e-mail address you used to create your Skype ID and then click on submit.
After doing this, your Skype ID will be auto-filled in the first field of the form.
2. You just have to log into Skype for web here:
https://web.skype.com/ and then click on your name and it will be available on the web page.
You will be logged into Skype for web (make sure that you don't try to use the SkyPrivate plugin on Skype web since it won't work).