Visualising $\pi_2(S^2)$ and $\pi_2(\mathbb{R}P^2)$

Question

I would like to know how to visualise the elements of $\pi_2(S^2)$ as unit vector fields on the sphere. For instance, the generator $a$ of $\pi_2(S^2)$ would be visualised as a 'hedgehog' configuration:

(To be clear, I'm considering the grey sphere as the $S^2$ domain, and the red arrows as elements of the $S^2$ image, in classifying maps $f:S^2 \to S^2$. Considered in this way, all homotopy groups could be visualised as fields on spheres.)

I would like, in particular, to know what the vector fields are corresponding to the $a^2$ and $a^{-1}$ elements of $\pi_2(S^2)$. To me, this seems like an easier method of visualising this particular homotopy group than trying to imagine a sphere wrapped around another sphere a certain number of times, with self-intersections allowed, as in this image:

Likewise, viewing elements of $\mathbb{R}P^2$ as undirected unit vector fields, I would like to know how to visualise elements of $\pi_2(\mathbb{R}P^2)$ as vector fields on the sphere. My guess is that here the generator looks much like the 'hedgehog' above (without the arrow tips), but I am less sure of this.

Thanks for the help.