So you have a cyclic group acting on complex affine space, and the situation at the level of function fields is clear enough: by basic Galois theory the extension of rational functions is a cyclic extension of degree n. It is a Kummer extension, even: write down a linear combination of the indeterminates with nth roots of unity as coefficients in a kind of obvious way, and you have an element whose nth power is cyclic, but transforms under a non-trivial character of the cyclic Galois group.
I think what you want is a description of the quotient as an affine variety. (Have to go now.)