Here is some standard machinery generalizing this result, which may be more elementary than the above references.

Assume $G$ acts strongly transitively on a thick building $\Delta$. Let $B$ be the stabilizer of the fundamental chamber, $N$ the stabilizer of the fundamental apartment, $T$ the subgroup fixing the fundamental apartment, and $W$ the quotient $N/T$. Then $(G,B,N)$ is a $BN$-pair, also called a Tits system, for $G$. In particular, you have the Bruhat decomposition $G=\coprod_{w\in W}BwB$, plus lots more.

In this case, take $G$ to be $GL(V)$, and take $\Delta$ to be the flag complex of subspaces of $V$. The stabilizer $B$ of the fundamental chamber is the upper-triangular matrices. The stabilizer $N$ of the fundamental apartment is the monomial matrices; the subgroup $T$ fixing the fundamental apartment is the diagonal matrices; and the Weyl group $W$ is the quotient $N/T$, which can be identified with the permutation matrices.

You can find all the above, including proofs, in Chapter V.2 of "Buildings" by Brown. For the special case of $GL(V)$, you could also look at Exercises 7 and 8 in Chapter 2.4 of "Groups and Representations" by Alperin-Bell.

Here is some standard machinery generalizing this result, which is more combinatorial than the standard reductive-groups proofs (but gives far less understanding than e.g. Matt Emerton's explanation above).

Assume $G$ acts strongly transitively on a thick building $\Delta$. Let $B$ be the stabilizer of the fundamental chamber, $N$ the stabilizer of the fundamental apartment, $T$ the subgroup fixing the fundamental apartment, and $W$ the quotient $N/T$. Then $(G,B,N)$ is a $BN$-pair, also called a Tits system, for $G$. In particular, you have the Bruhat decomposition $G=\coprod_{w\in W}BwB$, plus lots more.

In this case, take $G$ to be $GL(V)$, and take $\Delta$ to be the flag complex of subspaces of $V$. The stabilizer $B$ of the fundamental chamber is the upper-triangular matrices. The stabilizer $N$ of the fundamental apartment is the monomial matrices; the subgroup $T$ fixing the fundamental apartment is the diagonal matrices; and the Weyl group $W$ is the quotient $N/T$, which can be identified with the permutation matrices.

You can find all the above, including proofs, in Chapter V.2 of "Buildings" by Brown. For the special case of $GL(V)$, you could also look at Exercises 7 and 8 in Chapter 2.4 of "Groups and Representations" by Alperin-Bell.