I just got my Razr maxx. Tried Skype and I can see the person but I can't hear them and they can't see me. I can see them. I think skype does not work with Razr and my wifes HTC Rezound. She can hear, but the contact cannot see her.
I was having the same problem and neither Verizon or Skype was able to help. I also have a Lenovo Android Thinkpad Tablet and was having the same problem with that and Skype. Neither could make a Video call!
I happened to come across a web site with QR Bar Code that brought me to a site that had an older version of the Skype.APK. I downloaded it and installed it to my Tablet and found that Skype was working fine for both Audio and Video calls.
I copied that to DropBox and then installed it on my Droid RAZR MAXX. Viola' now both are working correctly with Skype. Even after the Google Play Store updated it, it's still working on both phone and tablet.
P.S. Neither Verizon or Skype technical support were able to help with this. I found it incredible that the Skype Support team are unable to place a video Skype call. If not them, then who?
If you want to try this email me at email@example.com and I'll send a link to my Dropbox for you to try it.