Second Lie algebra cohomology with values in $\mathbb{R}$ is finite dimensional- reference request

Question

While studying projective representations of Lie groups, I have come to the notion of Lie algebra cohomology. $H^2({\mathfrak{g},\mathbb{R}})$ is in bijective correspondence with equivalence classes of central extensions of $\mathfrak{g}$ by $\mathbb{R}$. I have come to the conclusion, that $H^2(\mathfrak{g},\mathbb{R})$ is an $\mathbb{R}$-vector space, as it is the quotient of two $\mathbb{R}$-vector spaces (which inherit pointwise addition and multiplication). I would like to conclude that for any finite-dimensional Lie algebra $\mathfrak{g}$ (need not be simple,semisimple or anything), there exists some $n \in \mathbb{N}$, such that as groups, or as vector spaces it holds that \begin{equation} H^2(\mathfrak{g},\mathbb{R}) \cong \mathbb{R}^{n}. \end{equation} In order to obtain this conclusion, I would need a statement/proof of the fact that for any finite dimensional Lie algebra $\mathfrak{g}$, its second cohomology group is finite dimensional. Then, upon the choice of a basis, I could show that it is isomorphic to some $\mathbb{R}^{n}$, as it is an $\mathbb{R}$-vector space. Where can I find such a reference, or how could I prove this?