Eigenvectors of $n^{th}$ root of identity matrix

Question

Let $A^n=I,$ where $A$ is $n\times n,$ and assume that $A^k\neq I,$ for all $1\leq k<n.$ Since its characteristic polynomial is $x^n-1$, the distinct $n^{th}$ roots of unity are its eigenvalues,

Thus there are a full set of linearly independent eigenvectors.

• What do they look like?
• If we assume $A$ is orthogonal, what do they look like?
• If we assume $A$ is real, can one say anything more?