A ring $R$ is uniformly strongly prime if some finite $S \subseteq R$ is such that for $a,b \in R$, $aSb = \{0\}$ implies $a$ or $b$ is zero, in which case the bound of uniform strong primeness of $R$ is the smallest possible size of such an $S$. The case of matrix rings $R$ is considered. Via vector multiplication and bilinear equations, we obtain alternative definitions of uniform strong primeness, together with new theorems restricting the bound of uniform strong primeness of these rings.