Matrix valued one form $ \text{tr}(C_1 C_2 \cdots C_{n+1}) = \frac{1}{(n+1)!} \sum_{\sigma \in S_{n+1}} tr(C_{\sigma_1} \cdots C_{\sigma_{n+1}}) $

Question

Let $C_i$ be a matrix valued one form, then I want to prove \begin{align} \operatorname{tr}(C_1 C_2 \cdots C_{n+1}) = \frac{1}{(n+1)!} \sum_{\sigma \in S_{n+1}} \operatorname{tr}(C_{\sigma_1} \cdots C_{\sigma_{n+1}}) \end{align} How can I prove this in general?

My former question was restricted to a matrix and it turned out to be wrong, So I give more constraint.