In order to understand representations of the general linear group $GL(n,K)$, with $K=\mathbb{R}$ or $K=\mathbb{C}$, I'm looking for an explicit construction of the polynomial representation matrices mapping $GL(n,K)$ to $GL(m,K)$. The paper Stanley; $GL(n,\mathbb{C})$ for combinatorialists, doi:10.1017/CBO9781107325548.009 gives the following example for the polynomial representation $\phi$ of $GL(2,K)$ with $m=3$
$ \left[\begin{matrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{matrix}\right] \quad \xrightarrow{\phi} \quad \left[\begin{matrix}a_{11}^{2} & 2 a_{11} a_{12} & a_{12}^{2}\\a_{11} a_{21} & a_{11} a_{22} + a_{12} a_{21} & a_{12} a_{22}\\a_{21}^{2} & 2 a_{21} a_{22} & a_{22}^{2}\end{matrix}\right] $
Sturmfels "Algorithms in Invariant Theory", doi:10.1007/978-3-211-77417-5 on page 138 lists the case for $m=4$
$ \left[\begin{matrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{matrix}\right] \quad \xrightarrow{\phi} \quad \left[\begin{matrix}a_{11}^{3} & a_{11}^{2} a_{12} & a_{11} a_{12}^{2} & a_{12}^{3}\\3 a_{11}^{2} a_{21} & a_{11}^{2} a_{22} + 2 a_{11} a_{12} a_{21} & 2 a_{11} a_{12} a_{22} + a_{12}^{2} a_{21} & 3 a_{12}^{2} a_{22}\\3 a_{11} a_{21}^{2} & 2 a_{11} a_{21} a_{22} + a_{12} a_{21}^{2} & a_{11} a_{22}^{2} + 2 a_{12} a_{21} a_{22} & 3 a_{12} a_{22}^{2}\\a_{21}^{3} & a_{21}^{2} a_{22} & a_{21} a_{22}^{2} & a_{22}^{3}\end{matrix}\right] $
I'd be most grateful for pointers to the literature providing an explicit construction for arbitrary $m$. Thanks.