Jelly, 12 bytes
*`ḶbµQ⁼$Ðfḅ³
Will work for any n, given enough memory, Jelly's base conversion is not restrictive.
How?
*`ḶbµQ⁼$Ðfḅ³ - Main link: n
µ - monadic chain separation
* - exponentiation with
` - repeated argument, i.e. n^n
Ḷ - lowered range, i.e. [0,1,2,...,n^n-1]
b - covert to base n (vectorises)
Ðf - filter keep:
$ - last two links as a monad
Q - unique elements
⁼ - equals input (no vectorisation)
³ - first program argument (n)
ḅ - convert from base (vectorises)