Skype on iPad won't stay online
Took me a lot of time to discover you have an option in iPhone (iPad) Settings app, you scroll down until you find Skype, and there you have the option to go offline automatically. Make sure this is set appropriately (e.g. to Never).

I'm running last version available today, 23rd of may 2012, both IOS and Skype.
Solved my problem few weeks ago.