Initially I was using a distance metric based on principle angles from this paper to calculate distances in the Grassmannian manifold. However, my project slightly changed and it seems like my manifold is now composed of ordered vectors. Hence I found that the Stiefel manifold might be more appropriate.
However, I don't know if distance metrics for the Grassmannian can still be used in my case. From here it seems like I need a different distance metric for the Stiefel, but I'm not sure where to find any.
A distance function on the Grasmannian $d_G:Gr(n,k)\times Gr(n,k)\to\mathbb{R}$ won't suffice, in that we cannot define a distance function on the Stiefel manifold $d_S:V(n,k)\times V(n,k)\to\mathbb{R}$ by $d_S(E,F):=d_G(\operatorname{span}(E),\operatorname{span}(F))$. This is because there are distinct $k$-frames $E\neq F$ which span the same subspace, and defining $d_S$ in this manner would give $d_S(E,F)=0$.
Still, there are many possible distance functions on $V(n,k)$, and which to choose depends on the properties you want this distance to have. Since $V(n,k)$ embeds canonically into $\mathbb{R}^{nk}$, one could simply use Euclidean distances in $\mathbb{R}^{nk}$, or alternately use the geodesic distance from the Riemannian metric induced by this embedding.