In this paper, a symplectic fission scheme for the association scheme of $m\times n$ rectangular matrices over the finite field ${\mathbb{F}}_q$, denoted by $\mathrm{S}\mathrm{M}\mathrm{a}\mathrm{t}(m\times n,q)$, is constructed, where $q$ is a power of a prime number. We discuss its association classes and inner automorphism group. In particular, we determine the intersection numbers and automorphism group of $\mathrm{S}\mathrm{M}\mathrm{a}\mathrm{t}(m\times n,q)$ for $m=1$ and $m=2$.