I know that the set of doubly stochastic matrices $(\Omega(n))$ form a polyhedron. I only know about polyhedron is that it is a $3$-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. Also, I read that
Birkhoff proof that set of all doubly stochastic matrices is a convex combination of permutation matrices.
I studied a statement form research article about doubly stochastic matrices is
doubly stochastic matrices $(\Omega(n))$ form a closed bounded convex polyhedron in Euclidean $n^2$ space whose dimension is $(n-1)^2$ and whose vertices are the $n\times n $ permutation matrices.
Can someone give advice on how to visualize this statement in $\mathbb{R}^{n^2}$? Or give some examples of this type polyhedron if exist? Also I am not getting that dimension part, namely, how it is $(n-1)^2$.
Any hint is appreciated.