What version of iOS are you running? If it's an older version, you might not be able to upgrade. If you're running a current version of iOS, then try doing a hard reset (hold the home and power button simultaneously until your phone cuts off and the silver apple logo reappears).
The App Store is running slower for me right now but I've searched for and found the Skype version 5.0 app listing. Maybe the issues are geographic in nature, with problem manifestations based on your location when trying to access services. It could be that the various Apple servers are not equally affected.
"The first indication that there was something wrong was when several users noted that they were unable to download the Skype app when it was released this morning. Some users received a message that the app was no longer available for download."