This is a general question that asks whether there is geometric significance to differential forms that restrict to G-invariant volume elements on the fibers of a principal G-bundle.

For an SO(2) bundle this is certainly true since the exterior derivative of such a form is the pull back of the Euler class of the bundle. In this case, the form itself is just a connection 1-form and is also a "global angular form".

The connection 1-form on an oriented circle bundle is a special case of a differential form that restricts to a G-invariant volume element on the fibers of a principal G-bundle. So do higher dimensional examples also contain topological significance? Do they generalize the idea of a global angular form?

Strangely for the tangent bundle, dimension 2 is the only dimension where the exterior derivative of this form on the tangent SO(n) bundle has the same dimension as the manifold. In higher dimensions, the dimension of the form is bigger than the dimension of the manifold. So for the tangent bundle one can not pull something back from the base except in dimension 2.

Still one might ask if such higher dimensional forms ever have geometric significance, perhaps even on the principal bundle itself. One might look at principal SO(3)or SU(2) bundles over 4 manifolds for instance or principal SO(4) bundles over 8 manifolds.

Some maths:In the context of symplectic topology, when you have a Hamiltonian fibration, i.e. a fibre bundle $M \to E \to B$ whose fibre is a symplectic manifold $(M, \omega)$ and the structure group is $Ham(M, \omega)$, you can construct a two-form $\Omega$ on the total space $E$ which restricts to $\omega$ in each fibre. This form is called acoupling form... $\endgroup$ – Oldřich Spáčil Mar 20 '15 at 17:10