Let $n \geq 1$ be an integer. We say that a matrix $M \in \mathcal{M}_n(\mathbb{R})$ is doubly stochastic if for all $i, j \in \{1, \ldots, n\}$ we have
$$M_{ij} \geq 0 \quad \text{and} \quad \sum_{k=1}^n M_{ik} = \sum_{k=1}^n M_{kj} = 1.$$
We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$.
Show that $B_n$ is a polytope and determine its dimension.