Identity regarding problem 2.4 of Walds General Relativity

I was going through some of the problems in Wald’s General Relativity and in problem 4 chapter 2 I found something that confuses me. So, basically we are asked to show that in any coordinate chart $(x^1,ldots,x^n)$ the following equation holds
begin{align}
Bigr(partial_nu Y^{gamma *}_mu – partial_mu Y^{gamma * }_nuBigr)dx^muotimes dx^nu &= Bigr(C_{alpha beta}^gamma Y^{alpha *}_mu Y^{beta *}_nuBigr)dx^muotimes dx^nu, tag{$1$}
end{align}
where ${Y_alpha}_{alpha=1}^n$ is a frame in a $n$ dimensional manifold, ${Y^{gamma *}}_{gamma=1}^n$ its dual frame and $C_{alphabeta}^gamma=bigr(Y^{gamma *}bigr)bigr[Y_{alpha},Y_{beta}bigr]$ the structure constant for the commutator. This can easily be shown to be the case by evaluating both sides of eq. 1 at $Y_rhootimes T_sigma$. In local coordinates the LHS of (1) takes the form
begin{align}
Bigr(partial_nu Y^{gamma *}_mu – partial_mu Y^{gamma * }_nuBigr)dx^muotimes dx^nuBigr(Y_rho^lambda partial_lambdaotimes Y_sigma^etapartial_etaBigr)&=Bigr(partial_nu Y^{gamma *}_mu – partial_mu Y^{gamma * }_nuBigr)Y_rho^mu Y_sigma^nu
&=-Y_mu^{gamma*}Y_sigma^nupartial_mu Y_rho^nu+ Y^{gamma *}_nu Y^{mu}_rhopartial_mu Y^nu_sigma
&=-Y^{gamma*}_mu bigr( Y_sigma^nu partial_nu Y_rho^mu – Y_rho^nu partial_nu Y_sigma^mu bigr)
&=Y^{gamma*}bigr[Y_rho,Y_sigmabigr]
&=C_{rho sigma}^gamma.
end{align}
Similarly, by evaluating the RHS we arrive at the same result. Since ${Y_rhootimes Y_sigma}_{rho,sigma=1}^n$ is a basis for the space of tensors of rank 2, this successfully proves the desired result. However, and this is what confuses me, we may have as well evualuated both sides of eq. 1 with another basis, namely ${partial/partial x^rhootimespartial/partial x^sigma}_{rho,sigma=1}^n$. In which case the LHS becomes $
partial_sigma Y_rho^{gamma *}-partial_rho Y_sigma^{gamma *}$; where as the RHS can be written as
begin{align}
C_{alpha beta}^gamma Y^{alpha *}_rho Y^{beta *}_sigma&=Bigr(partial_nu Y^{gamma *}_mu – partial_mu Y^{gamma * }_nuBigr)Y_alpha^mu Y_beta^nu Y^{alpha *}_rho Y^{beta *}_sigma.
end{align}
So this seems to imply the relation
begin{align}
Y_alpha^mu Y^{alpha *}_rho &= delta^mu_rho,
end{align}
which seems reasonable, but I do not quite understand it. Is this relation true for any frame ${Y_alpha}$? If so, how does one prove it?