Bruhat decomposition of flag variety.

Question

Let $GL_{n}/B_{n}$ be the complete flag variety (where $B_{n}$ is the group of upper-triangular nonsingular matrices). Let $W$ be the Weyl group of $U(n)$ (that in this contest is a permutation group). We have $GL_{n}/B_{n} \simeq U(n)/T_{n}$ (where $U(n)$ is the unitary group and $T_{n}$ its maximal torus). I have to build an explicit Bruhat decomposition of $U(n)/T_{n}$. Could you help me to find the cells end prove that there are $|W|$ cells of even dimension? (Also in a particular situation, for example $U(3)/(S^{1})^{3}$.)