Consider a vector space with a linear symplectic form $(V, \Omega)$. I am trying to see that the following sequence is exact:
$$0 \to S^3 V^* \xrightarrow{i} \text{Hom}(V, \mathfrak{gl}(V, \Omega)) \xrightarrow{\partial} \text{Hom}(\Lambda^2 V, V) \xrightarrow{\partial_\Omega} \Lambda^3 V^* \to 0$$
Here, for $p \in S^3 V^*$, the element $i(p)$ is such that, for $u, v \in V$, we have $\Omega(i(p)(u)(v), w) = p(u, v, w)$. As for the other maps, given $f \in \text{Hom}(V, \mathfrak{gl}(V, \Omega))$ we set $$(\partial f)(u, v) = f(u)(v) - f(v)(u)$$ and for $\phi \in \text{Hom}(\Lambda^2 V, V)$ we set $$(\partial_\Omega \phi)(u, v, w) = \Omega(\phi(u, v), w) + \Omega(\phi(v, w), u) + \Omega(\phi(w, u), v)$$
I calculated that all compositions are zero and I can also see that the sequence is exact at the $i$ map, but I am stuck as for the other ones. I tried mimicking the calculation of the $\mathfrak{o}(n)$ case, but since in this case the kernel is not zero, the choice is not unique and my calculations lead nowhere.