Definitions. Given an oriented vector space $V$ with metric $g$, a linear $k$-calibration on $V$ is a $k$-form $\rho\in\Lambda^k V^*$, such that for every oriented $k$-dimensional subspace $W\subset V$, $\rho\vert_W\leq\sigma_W$, where $\sigma_W$ denoted the volume form induced by the metric $g$ on $W$. The inequality makes sense because the orientation of $W$ determines a positive direction on the real line $\Lambda^kW$.
Let $(V,g,I,\omega)$ be a vector space with compatible symplectic, complex structures and metric. I want to show that $\frac{1}{k!}\omega^k$ is a linear $2k$-calibration on $(V,g)$.
My attempt. Let $W\leq V$ be an oriented space of dimension $2k$. By Darboux's theorem we can find a $\{e_i,f_i\}_i$ for $V$ such that $$ \omega=\sum_i e_i^*\wedge f_i^*. $$ Then I believe $$ \frac 1{k!}\omega^k=\sum_{i_1<\dots<i_k}e^*_{i_1}\wedge f^*_{e_1}\wedge\dots\wedge e^*_{i_k}\wedge f^*_{i_k}. $$ Let $\sigma_W$ be the volume form induced by the metric $g$ on $W$. Given an orthonomal basis $\eta_i$ for $W$, we can write $$ \sigma_W=\eta_1\wedge\dots\wedge \eta_{2k}. $$ How can I relate these those forms to get the desired inequality $\frac 1{k!}\omega^k\leq \sigma_W$?
Edit. Maybe I should try induction, will do that today.
