Let $\mathbb{F}$ be a finite field. Let $A\le M_n(\mathbb{F})$ be a matrix algebra.

Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$ that one needs to take such that, with high probability, the algebra generated by $a_1,\dots,a_k$ is $A$?

I am interested in a bound that applies to all $A$. Replacing "high probability" by "probability bounded away from zero" would also be fine.

Taking $k=O(n^2)$ will provide a set that spans $A$ as a vector space. Would $O(n)$ suffice? $O(\log n)$?