Random pairings with no repeats for a whole semester

I have classes of 25-35 university esl students and I would like to find a way to pair up students randomly--but with no repeats. Each student would get a new partner every time throughout the whole 15-week semester. Students would only be paired with someone they've already worked with when they've already worked with every other student in the class or other factors mandate repeat pairings.

So anything that would enable me to do that would not only have to be able to pair students at random but would also have to remember which students have been paired with each other to prevent repeats over the course of the whole semester. It's hard because it has to deal quirks like absent students, etc.

There are plenty of “random team generators” on the internet, but I couldn’t find any that solve the problem of repeats over the course of a whole semester. I have seen some complicated suggestions using Excel, but none that I could find really matched my specific needs.

Also, I am an extreme novice using Excel.

Thanks Kaper. I'm an excel novice, so it's taken a bit of time and effort to try out the round robin method. But I think it just might work. Thank you very much!