If $\mathfrak{g}\subseteq\mathfrak{so}(V)$, then $H^{0,2}(\mathfrak{g})=(\mathfrak{so}(V)/\mathfrak{g})\otimes V^*$ Problem
I'm learning about $G$-structures and was assigned this exercise (Cartan for Beginners Exercise 8.3.6.1):

Let
  $$
H^{0,2}(\mathfrak{g})=(V\otimes\Lambda^2V^*)/\delta (\mathfrak{g}\otimes V^*),
$$
  where $\delta:(V\otimes V^*)\otimes V^*\to V\otimes\Lambda^2V^*$ is the skew symmetrization map.
  If $\mathfrak{g}\subseteq\mathfrak{so}(V)$, then $H^{0,2}(\mathfrak{g})=(\mathfrak{so}(V)/\mathfrak{g})\otimes V^*$.

I'm trying to figure this out, but I'm not very comfortable with tensors, let alone my ability to swim through isomorphisms etc., so I'm stuck. I figure I will do a classic $A\subseteq B$ and $B\subseteq A$ implies $A=B$ argument, but I'm having trouble figuring how an element $X\in\mathfrak{g}$ will decompose into the form $X=a^i_jv_i\otimes v^j$ order to even work with $X\otimes V^*=a^i_{jk}v_i\otimes v^j\otimes v^k$ and hence $\delta(\mathfrak{g}\otimes V^*)=a^i_{jk}v_i\otimes v^j\wedge v^k$.
Please let me know if this needs clarification; I'm not very sure which notations are standard and which are created by the author. I've looked around on the internet but couldn't find anything. Any help is appreciated. Thanks in advance.
 A: If you feel uncomfortable with tensors, you can do everything in the setting of linear maps. Starting from a linear map $A:V\to L(V,V)$, you obtain a skew symmetric, bilinear map $\delta A:V\times V\to V$ via $\delta A(v,w)=A(v)(w)-A(w)(v)$. Now if you have a linear subspace $\mathfrak g\subset L(V,V)$, you can look at the restriction of $\delta$ to $L(V,\mathfrak g)\subset L(V,L(V,V))$. Denoting by $L^2_a(V,V)$ the spaces of skew symmetric bilinear maps $V\times V\to V$, you want to determine the quotient $L^2_a(V,V)/\delta(L(V,\mathfrak g))$. 
To solve your problem you can proceed as suggested in the comment by @mt_ . First consider the case that $\mathfrak g$ is the space $\mathfrak{so}(V)$ of maps $V\to V$ which are skew symmetric with respect to a given inner product on $V$. Determine the dimensions of the space $L(V,\mathfrak{so}(V))$ and $L^2_a(V,V)$ and directly prove injectivity of $\delta$ by computing its kernel. Thus $\delta$ is an isomorphism in the case of $\mathfrak{so}(V)$, which easily implies the claimed result for any $\mathfrak g\subset\mathfrak{so}(V)$. 
