We know that $ֿ\pi_2(\mathbb{S}^2)\cong\mathbb{Z}$. Given a fixed degree, say $n\in\mathbb{Z}$, is there a standard, conventional way to parametrize continuous maps $\mathbb{S}^2\to\mathbb{S}^2$ which have degree $n$?
It is pretty clear that the identity map (in spherical coordinates) $(\theta,\varphi)\mapsto(\theta,\varphi)$ generates the group and furthermore, $$(\theta,\varphi)\mapsto(n\theta,\varphi)$$ yields a representatative in the homotopy class of degree $n$. Is there some way of starting off from that map and somehow enumerating all possible deformations of that map which keep you in the same homotopy class? For example, it is clear that one of these parameters is simply a choice of member of $SO(3)$ which picks the initial orientation of the map above.
Ideally I would like a decomposition (into a sum of two functions, or a composition of two functions), one of which has degree $n$ and is somehow special to that homotopy class and the other has degree zero, and enumerating it is the same for all $n$.
Does something like that exist already?