This is an unfruitful search for a numerical relation between subspaces of a Euclidean space. I now think that the relation between two subspaces B and C of an n dimensional vector space with positive definite quadratic form is entirely captured by the shape of the following body: Take the unit sphere in B (or C), orthogonally project it into the other sub-space, and then orthogonally project it back. The result will be an ellipsoid whose shape (in turn captured by the lengths of its axes), captures the metric relationship between the two subspaces.

The eigenvalues of these double projections are all between 0 and 1 inclusive. If such eigenvalues are all 0 then the spaces are orthogonal. If they are all 1 then one space is included in the other. These eigenvalues (or axis lengths) are cosines of the angles between the two spaces in a sense defined hereby.

By ‘capturing the numerical relation’ I mean that for two configurations yielding congruent doubly projected bodies, a rigid rotation of the embedding space can bring one configuration to the other; the first pair of subspaces is congruent to the second pair.

The following stuff goes much beyond what is here:

Differential Geometry of Grassmann Manifolds

Riemannian geometry of Grassmann manifolds with a view on algorithmic computation

An Introduction to Finsler Geometry

On the geometry of complex Grassmann manifold, its noncompact dual and coherent states

See this for representing Grassmann values numerically.