You can compute solid angle (a.k.a. spherical excess, normalized spherical surface area) by taking a surface integral, but the units are not "square degrees" or "square radians", but instead an entirely new type, usually just measured in radians ("steradians"). Some people have defined https://en.wikipedia.org/wiki/Square_degree but that is a stupid unit.
While rotation is naturally oriented like a bivector (plane), solid angle is naturally oriented like a trivector (3-space).
The natural representation is as a kind of (unitless) ratio formed from 3 vectors, not the product of two vector–vector ratios.
Curious what you think of the d_theta * d_phi term in a spherical coordinates integral...