Let $V$ be a finite-dimensional Hilbert space and $Gr_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$. The $k$th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by extending $$ \langle v_1\wedge \cdots \wedge v_k, w_1\wedge \cdots \wedge w_k\rangle_{\bigwedge^k(V)} := \det{\left(\langle v_i,w_j\rangle_{V}\,_{i,j}\right)} $$ bilinearly to all $k$-vectors.
The Grassmannian can be equipped with a metric which measures the angle (more exactly I think the sine of the angle) between two $k$-dimensional subspaces by $$ d(U,W) := \left\| P_U-P_W \right\| $$ where $P_U:V\to U$ is the projection onto U and the norm is the operator norm (see for example this discussion and Equation (3), p. 3428 in the article linked therein).
I know that there is the Plücker embedding which maps $G_k(V)$ to the projectivisation of the $k$th exterior power. In fact, the image of the Plücker embedding corresponds to the simple $k$-vectors.
For example, if $U$ and $W$ coincide then the $k$-wedge of two distinct bases are off by a scalar factor (the determinant of the change of basis matrix). So it would be nice to have something like $$ d(U,W) \leq \left\|u_1\wedge\cdots\wedge u_k - w_1\wedge\cdots\wedge w_k\right\|_{\bigwedge^k(V)} $$ for any bases $(u_i)$ and $(w_i)$ of $U$ and $W$.
Is there a way of estimating the angle metric on the Grassmannian in terms of the norm on the exterior power?