In your Table you should use E to get the exponential function.
And you may want to have a look at this
n = 5;
mat = Table[(E^(2 I Pi/n))^(i*j), {i, 0, n - 1}, {j, 0, n - 1}];
dd = Det[mat]
dd1 = dd /. a_. Exp[b_ Complex[0, x_] ] -> a (Cos[x b] + Im Sin[x b])
dd1 // Expand
dd // FullSimplify
