There is no skype name for the new skype accounts...

Sort of... There is one but it's hidden, auto-generated and it isn't used for login.

To find your skype name:

• Open Skype's Preferences -> Messaging;
• Look for a setting named "Identify SMS Messages by:";
• There should be an option "My Skype Name (live:...).

Everything withing the parentheses is your skype name "live:..." etc.

I first decided to look for my skype name by searching for my account in Add Contact, but there I get a different string which doesn't work.

Once you have the name...

You can use skype's Contact me button generator

Which is located, here.

Or you can build your own

For call html links (replace "live..." with your skype name):

<a href="skype:live:...?call" target="_blank">
    Call me
</a>

For add html links (replace "live..." with your skype name):

<a href="skype:live:...?add" target="_blank">
    Contact me
</a>

Thanks to

Hristo Kosev (you can follow him on twitter @hkosev) who helped me figure this out.

Cheers!