In my opinion this will boil down to following three cases. If you agree with this, and your case falls into one of the three clases, please give it a thumb.

1. BT device does not work at all because Handsfree (HFP) profile is not supported by both
   The BT and iOS devices must BOTH support Handsfree profile to be able to feed mic input to Skype (or any other iOS app). Whether your Apple device can do this is listed here: iOS: Supported Bluetooth profiles and about the handset, see its technical specification. (Non-technically: You have a bluetooth device that works for normal calls via cell phone and you are trying to make Skype calls using iPad 1st gen, or older iPod Touch (2nd, 3rd gen).)
2. BT device only works for audio output (you hear them, but mic does not work)
   Case when both your BT and Apple device support A2DP profile
3. Both devices support Bluetooth Handsfree profile and only first call works. That's the case we are all ranting about. In this case, the BT device can be used to make cell calls and works flawlessly with many other applications, but due to bug in Skype (presumably) it only works with Skype for the first call. And this is on for like seven months, tens of people are reporting and Skype does not pay any attention.

I've tried to elevate this issue before, see support case id: 850107. Specifically, I have iPad 3, 5.1, Sony Ericsson BT VH110, Skype 4.0.1264 (and just the first call works)