I am trying to understand Bruhat decomposition in practique, and more generally matrices decompositions. Let me take the example of $G=GL(3)$, and introduce $B$ the subgroup of upper triangular matrices, $N$ of unipotent upper triangular matrices, and $D$ of diagonal ones. The Bruhat decomposition is $$(1) \qquad G = \bigcup_w BwB$$
where elements $w$ are running through the permutation group in $GL(3)$. I would like to write such a decomposition not with $B$,but with $N$, that is to say compute $N \backslash G / N$. For this I tried to do so for every classes appearing in $(1)$, for instance the trivial one gives
$$B1B = B = NDN$$
However for every other permutation matrices I am lost: should I blindly try, like with general coefficients and doing computations explicitly, or is there a more systematic method?