I need a reference for the following statement:

Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $k[G]^n$ for some $n$ where $k[G]$ is coordinate ring of $G.$

I could find this statement in Steinberg's lecture notes on "Conjugacy Classes in Algebraic groups" but am not happy with the proof there.

Thanks in advance.