Let $K_{n,n}$ be a complete bipartite graph with two parts $\{u_1,u_2,\ldots,u_n\}$ and $\{v_1,v_2,\ldots,v_n\}$, and let $K^-_{n,n}$ be the graph derived from $K_{n,n}$ by delete a perfect matching $\{u_1v_1,u_2v_2,\ldots,u_nv_n\}$.

Since $K^-_{n,n}$ is now $(n-1)$-regular, it has $n-1$ disjoint perfect matchings. My question is whether the edges of $K^-_{n,n}$ with $n\geq 4$ can be decomposed into $n-1$ disjoint perfect matchings in such a way that in each matching $M$, if $u_iv_j\in E(M)$ then $v_iu_j\not\in E(M)$.