You have to do this through the People app now. Find your Skype contact and tap/select it. Then bring up the menu bar by swiping up or right clicking. Select Edit and make your changes. You can save by using the same menu bar.
One thing to be careful of is if you have your Skype contact name linked to other accounts. I found you have to unlink your Skype account so it's by itself, make your changes, then you can re-link it if you'd like.