Consider a map $f:S^1\to U(1)=S^1$, since we know $\pi_1(S^1)=\mathbb{Z}$, which measures how many times the map "wind" around the circle. Given some explicit form of the function $f(\phi$), where $\phi$ is the coordinate on the circle, we can calculate the winding number by $$ \int_0^{2\pi}\frac{i}{2\pi f(\phi)}f'(\phi)d\phi$$
Given a map $f:S^2 \to S^2$ and it is known that $\pi_2(S^2)=\mathbb{Z}$. What is the quantity/formula being used to calculate the degree of the map?
In general, how does one construct the formula for calculating the winding number for arbitary $\pi_n(\mathcal{M})$ ?