Let $\Sigma_k$ be the $k$-th symmetric group and $B\Sigma_k$ be its classifying space. How to prove:

for any $n\geq 1$ and the $n$-skeleton $sk_n (B\Sigma_k)$, there exists a finite dimensional $CW$-complex $K$ such that

(i). $sk_n(B\Sigma_k)\subseteq K\subseteq B\Sigma_k$;

(ii). $H^*(K;\mathbb{Q})$ is trivial?

Could I just let $K=B\Sigma_k$?