Identity regarding the components of a dual basis

Question

This problem is from Robert Wald's "General Relativity." The problem is 4(b) from chapter 2. Let $Y_1\cdots Y_n$ be smooth vector fields on an $n$-dimensional manifold $M$ such that at each $p\in M$ they form a basis of the tangent space $V_p$. Then we expand the Lie bracket in this basis. $$ [Y_\alpha,Y_\beta]=C^{\gamma}_{~~~ \alpha\beta}Y_{\gamma}$$ Where $C^{\gamma}_{~~~ \alpha\beta}=-C^{\gamma}_{~~~ \beta\alpha}$. Now, let $Y^{1^*}\cdots Y^{n^*}$ be the dual basis. I am asked to show that the components of the dual basis satisfy the following identity. $$\partial_\nu (Y^{\gamma^*})_\mu-\partial_\mu (Y^{\gamma^*})_\nu = C^{\gamma}_{~~~ \alpha\beta}(Y^{\alpha^*})_\mu(Y^{\beta^*})_\nu$$ The hint to contract both sides of the equation with $(Y_\sigma)^\mu(Y_\rho)^\nu$ was given, and I did it. This just left me with the identity $$C^{\gamma}_{~~~ \sigma\rho} = \left[ \partial_\nu (Y^{\gamma^*})_\mu-\partial_\mu (Y^{\gamma^*})_\nu\right] (Y_\sigma)^\mu(Y_\rho)^\nu. $$ However, I am stuck after this short step. I've tried messing around with that identity and haven't gotten anywhere with it. Could someone give some hints?