I've been trying to prove that if $(M, J, g)$ is an hermitian manifold of real dimension $2n$ with Kähler form $\omega$, then $$\dfrac{\omega^n}{n!}=\mathrm{vol}_g,$$ where $\mathrm{vol}_g$ is the riemannian volume form of $(M, g)$.
Now, I know very well how this is usually done (please don't point that out, I don't want the usual proof using complex or real coordinates). The point here is that I'm trying to prove this in a very specific Riemanian way, namely: if $(E_1,\ldots,E_{2n})$ is an arbitrary local $g$-orthonormal frame (with no relation to $J$ whatsoever), it is easy to see that $$ \omega=\frac{1}{2}\sum_{k=1}^{2n}E_k^\flat\wedge (J E_k)^\flat .$$ So, in order to prove that $\dfrac{\omega^n}{n!}=\mathrm{vol}_g$, I just have to prove that $\dfrac{\omega^n}{n!} (e_1,\ldots,e_{2n})=1$ for any local $g$-orthonormal frame $(e_1,\ldots,e_{2n})$ (not neccesarily the same as $(E_1,\ldots,E_{2n})$, but we can use suppose $(e_1,\ldots,e_{2n})$ is $(E_1,\ldots,E_{2n})$ or $(JE_1,\ldots,JE_{2n})$ for that matter).
So, here comes the trouble. In order to compute $\dfrac{\omega^n}{n!} (e_1,\ldots,e_{2n})$, I can either try to get a nice expression for $\dfrac{\omega^n}{n!}$ first, or I can plug use the usual formula for the wedge product, but I haven't had any luck with either of them. As for the second option, I arrive at something like this $$\dfrac{\omega^n}{n!} (e_1,\ldots,e_{2n})=\frac{1}{2^n n!}\sum_{\sigma \in S_{2n}}(-1)^\sigma g(Je_{\sigma(1)}, e_{\sigma(2)})\cdots g(Je_{\sigma(2n-1)}, e_{\sigma(2n)}),$$ but I can't get anything useful out of that.
Right now I'm trying to get a compact expression for $\dfrac{\omega^n}{n!}$, but I haven't come to anything useful yet. If we write $\omega=\dfrac{1}{2}\sum\limits_{k=1}^{2n} \alpha_k$, then $\omega^n=\frac{1}{2^n} (\alpha_1 + \dots + \alpha_{2n}) \wedge \dots \wedge (\alpha_1 + \dots + \alpha_{2n})$, but I can't get a nice expression out of that (it is a $n$-fold wedge product of a sum of $2n$ two-forms) [please note it is not the same as the usual computation of the $n$-fold wedge product of a standard symplectic form in $\mathbb R^{2n}$].
I will aprreciate any hint or solution on how to handle this calculation. Regards