To atone for my misguided earlier attempt at an answer, I'd suggest a different way to get a general criterion of this sort (working first over an algebraically closed field, then adapting to finite fields via the BN-pair structure). While this is less elementary than the matrix theory over rings involved in the treatment of general linear groups, it explains the result conceptually in terms of the Bruhat decomposition. (Recall that this decomposition is a natural generalization to reductive groups of Gauss reduction in elementary linear algebra.)
The ideas here all go back to the 1956-58 Chevalley seminar, but the three textbooks Linear Algebraic Groups by Borel, Springer, and myself may be easier to refer to. In a connected semisimple (or reductive) algebraic group $G$, the Borel subgroups are all conjugate, so we can fix a particular $B$. Working in $G$ or in the (projective) flag variety $G/B$, we get a disjoint decomposition indexed by elements of the Weyl group $W$, starting with $G= \bigcup_w BwB$.
In the resulting Bruhat cell decomposition of $G/B$, the cell corresponding to $w \in W$ is isomorphic to an affine space of dimension $\ell(w)$. Then the longest element $w_\circ$ of $W$ (of length $= \dim G/B$, the number of positive roots) yields a dense open subset. The double coset $Bw_\circ B$ is similarly a dense open subset of $G$. Its left translate $\Omega:=w_\circ B w_\circ B$ is often called the big cell in $G$; it is in fact a principal open set in $G$.
Relative to a fixed maximal torus lying in $B$, we have $B = TU$ with $U$ maximal unipotent. Similarly $B^- := w_\circ B w_\circ = TU^-$. (Note that $w_\circ$ is its own inverse.) Here $U, U^-$ would be the upper and lower triangular unipotent matrices in the general linear group. Now recall a trivial lemma (7.4 in my book): the product of two nonempty Zariski-open (hence dense) subsets in $G$ is all of $G$. Apply this to $Bw_\circ B$, along with the fact that $T$ can be moved past other factors to combine with $B$, to conclude: $$G = (B w_\circ B) (B w_\circ B) = B w_\circ B w_\circ B = U U^- B.$$