On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the two people to choose said problem. I gave a very long convoluted argument which although correct was really inelegant. I proved it more than once because I wasn't satisfied with my proof, and I figured out a somewhat slick contradiction argument based on maximizing leading zeroes of rows (number of zeroes before the pivot), but the proof was still a real mess.

Statement of the problem:

Let $G:=GL(V)$ for $V$ a finite dimensional $k$ vector space. Let $B$ be the stabilizer of the standard flag (these will be invertible upper triangular matrices), and let $W$ be the subgroup of permutation matrices.

Show that $G=\coprod_{w\in W} BwB$, where the $BwB$ are double cosets. That is, show that $G=BWB$.

Question: Is there a slick proof of this fact maybe using "more machinery"? In particular, is there any sort of "coordinate-free" proof (Can we even define the Borel and Weyl subgroups without coordinates?)?