Is there any other date where/when I can write the exam?
Should I use when or where in that sentence
Unlike the people who have commented, I find only "when" to be normal here, as it is a date.
If I heard it with "where", I would assume the speaker was thinking of a 'date' not as a date on the calendar, but a place and time - which makes some sort of sense for an exam, which presumably has assigned places and times. But I would still find it odd.
Only 'when' makes sense to me, particularly as you specify "date". I mean, I'd know what you meant either way, but "where" would seem wrong - just an understandable sort of wrong.
Also, in English, we say "sit an exam" or "take an exam", not "write an exam".