Below, $x=\phi$ when $n=2$:
($\phi$ being the golden ratio)
Is there a way to express $x$ in terms of $\phi$ for $n>2$?
When $n=5$, we are talking about the roots of $x^5-x^4-x^3-x^2-x-1=0$. I suspect that, like most polynomials of degree 5, this one has Galois group $S_5$. If that's the case, then these roots can't be expressed in terms of the four arithmetical operations and square roots, cube roots, fifth roots, etc. It would follow that there's no expression in terms of the golden ratio (and arithmetic operations, and square roots, etc.).
It doesn't get any better for $n\gt5$.