Dimension of the best–worst choice polytope

For $n\ge 2$, let $[n]=\{1,\ldots,n\}$ be a set of alternatives. For each linear ranking $\pi=(\pi(1),\ldots,\pi(n))$ of $[n]$, define its deterministic best-worst choice vector $v_\pi\in\mathbb{R}^{n(n-1)}$, indexed by ordered pairs $(i,j)$ with $i,j\in[n]$ and $i\ne j$, by $(v_\pi)_{ij}=1$ if $\pi(1)=i$ and $\pi(n)=j$, and $(v_\pi)_{ij}=0$ otherwise. The best-worst choice polytope is $P_n=\operatorname{conv}\{v_\pi:\pi\text{ is a linear ranking of }[n]\}$. Determine an explicit formula, as a function of $n$, for $\dim P_n$.