From your N82 phone, go to OVI store and download/install skype from there. It works.
I had experienced the same problem like you had when attempting to install from skype.com/m or SMS. I also tried to download it from Nokia Suite on my PC, it did not work. Only the download from OVI store works. Hope it helps!
I am having problem downloading skype on my N82. I have the same problem. So i tried downloading from other sites. but the only thing that worked on my is the beta. and the beta sucks!! i cant use the free video call or even the regular call.. pls help me