Just confirming Alyssa'a answer: make-up test/exam in the US.

In the States I would call it a "make up exam" or "make up test". 

"The professor is letting me make up the test/exam"

"The professor is letting me redo the test/exam"

"The professor is letting me retake the test/exam"

"Oh, the make up exam was easier."

"I got to make up that exam, because I failed it the first time."

I would only use "to write an exam" if I am the one creating the exam questions. Otherwise, "to take an exam" or, more rarely, "to sit for the [named] exam." The named exams, for me using that verb, would be professional exams, like the bar (law test that lets you practice law) or the CPA exam (accounting).

@Chen 'Write' is not used in that way in the UK and I am surprised it is used that way in Canada. It sounds quite strange to me and I've never heard it used that way before. I'd be interested to hear if they use 'to write' that way in the US too, because in the UK we don't use it in that situation. Here, as you thought, 'write' would only be used for the person (the tEscher or professor) who sets the exam for the students

Here we say a person sits/takes an exam. 

If they have to take an exam again then they resit/retake the exam.

Wow, how interesting.

OK, so now I have concluded, beyond any reasonable doubt, that English is definitely not uniform throughout the globe.

So, with that in mind, I have decided that I should not move to another English speaking country, lest I be compelled to re-learn the little English I believe I already know :) :)

Great question Anivlam ...

Yes, this is one area where usage varies considerably from one English-speaking region to the next.

I had never heard the term 'make-up exam' before. Thank you for letting us know about that one, Alyssia. I would have presumed that it was some kind of practical assessment for beauticians.. hairdressing exam, make-up exam, lipstick, eye-shadow, that kind of thing :)