Let V be the irreducible $sl_{4}$-module with highest weight $\pi_{2}=\lambda_{1}+\lambda_{2}$ (i.e if $H=\left(\lambda_{1},\dots,\lambda_{4}\right)$ is a diagonal matrix in $sl_{4}$ with values $\lambda_{1},\dots,\lambda_{4}$ on it's diagonal then $\pi_{2}\left(H\right)=\lambda_{1}+\lambda_{2}$). Show that $\dim V=6$ and that there is a non-degenrate invariant quadratic form on $V$. Use this to create an Isomorphism of $sl_{4}$ onto the Lie-algebra of the orthogonal group in $6$ variables.
I know how to show that $V\simeq\wedge^{2}k^{4}$ and that it's dimension is $6$. I don't know how to find this quadratic form however. Thanks!