Two models of $BT^n$

Question

Let $T^n$ be the $n$-torus. We have (at least) two nice models of the classifying space $BT^n$:

1. $(\mathbb{CP}^\infty)^n$;
2. The flag manifold $F_n(\mathbb{C}^\infty)=V_n(\mathbb{C}^\infty)/T^n$, where $V_n(\mathbb{C}^\infty)$ is the Stiefel manifold of orthonormal $n$-frames in $\mathbb{C}^\infty$.

As the inclusion $V_n(\mathbb{C}^\infty)\subset (S^\infty)^n$ is $T^n$-equivariant, it induces an inclusion $F_n(\mathbb{C}^\infty)\subset(\mathbb{CP}^\infty)^n$. The universal property of classifying spaces show that this inclusion is a homotopy equivalence. As it is also a (closed Hurewicz) cofibration, it follows that $F_n(\mathbb{C}^\infty)$ is a deformation retract of $(\mathbb{CP}^\infty)^n$. However, I don't see how $(\mathbb{CP}^\infty)^n$ even retracts onto $F_n(\mathbb{C}^\infty)$.

So my question is: Is there a nice description of a deformation retraction?

Any help is appreciated. Thanks in advance!