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.