Is there a rotation representation that can also represent "turns", instead of collapsing coincident rotations into the same representation?

In 2D, a simple angle satisfies this, as it can have additional multiples of $2\pi$. For example, rotating by a turn and a half would be $3\pi$.

Is there something similar for 3D rotations? Does the concept even make sense there? Quaternions don't work for this since they only have two representations of any given rotation. Rotation vectors ($\theta\hat{e}$) seem to work, though they are very hard to work with.

EDIT: My objective with this is to extend quaternion spherical interpolation to rotations of more than 180° in terms of beginning and end "orientation with turns" objects, so you could, for example, interpolate over an entire revolution using the same machinery as you would with normal small rotation interpolation.