Non-trivial real valued two-cocycle on $\mathbb{R}^2$

Question

Assume that $c$ is an arbitrary non-vanishing antisymmetric bilinear form on $\mathbb{R}^2$ and view $\mathbb{R}^2$ as an Abelian Lie algebra. Can $c$ define a non-trivial real valued two-cocycle on $\mathbb{R}^2$?

Answer 1

By definition of the Chevalley Eilenberg cohomology, the space of $2$-cocycles with trivial coefficients $K$ is given by

$$ Z^2(L,K)=\{\omega\in {\rm Alt}(L\times L,K)\mid \omega([x_1,x_2],x_3)-\omega([x_1,x_3],x_2)+\omega([x_2,x_3],x_1)=0\}. $$

Of course, if $L$ is abelian, we obtain all antisymmetric (alternating) bilinear forms.