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Recall that for any inner product space $(\Bbb W, g)$, we get a natural $SO(g)$-module isomorphism $$\Phi: \mathfrak{so}(g) \cong \Bbb \Lambda^2 \Bbb W^*$$ given by identifying $\mathfrak{so}(g)$ as a subalgebra of $\mathfrak{gl}(\Bbb W) \cong \Bbb W \otimes \Bbb W^*$ and lowering an index using the inner product $g$; explicitly, this isomorphism is ...



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