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?)?

Edit: I take issue with Kevin Buzzard's comment mainly for the last line.  It makes me look like a jerk and a hypocrite, but that's not a fair characterization.  First of all, I've already proven this.  I was asking for a better proof.  That's more "due diligence" than any question I've voted down for being in a standard reference can claim.  Additionally, how can you honestly expect someone who's never seen the term "reductive group" before to be familiar with the references?  I searched for something like proof of Bruhat Decomposition of GL_n on Google, Google books, and Wikipedia and was not able to find a reference.  Granted, the last time I searched for it was right after the exam (probably a little less than a year ago), but that still doesn't mean that I failed to fulfill the necessary prerequisites for asking a question on MO.