Orbit space of usual action on n-cube $[0,1]^n$ by n-symmetric group $S_n$ is homeomorphic to topological n-simplex

Question

I do see it's true for $n=2$ i.e. action of $S_2$ on $[0,1]^2$ as it's just given by identification through reflection w.r.t. the diagonal $x=y$. But, I don't have a concrete map for higher $n$ 's.