Let $m,n\in\mathbb N$. Let $\{e_{ij}:1\le i,j\le m\}$ be the standard matrix units of $M_m(\mathbb C)$, let $\{f_{kl}:1\le k,l\le n\}$ be the standard matrix units of $M_n(\mathbb C)$, and let $\{E_{(i,k),(j,l)}:1\le i,j\le m,\ 1\le k,l\le n\}$ denote the matrix units of $M_{mn}(\mathbb C)$ indexed by the lexicographically ordered basis of $\mathbb C^m\otimes \mathbb C^n$. Then there is a unital $*$-isomorphism

\begin{align*} \Phi:M_m(\mathbb C)\otimes_{\min}M_n(\mathbb C)\to M_{mn}(\mathbb C) \end{align*}

determined by

\begin{align*} \Phi(e_{ij}\otimes f_{kl})=E_{(i,k),(j,l)} \end{align*}

for all $1\le i,j\le m$ and $1\le k,l\le n$.