As I have known, we can write down an explicit expression for $\pi_n(S^n)=\mathbb{Z}$, taking $\pi_2(S^2)$ as an example:
Consider a smooth map: $\mathbb{R}^2\rightarrow S^2$ that assign $x\mapsto \vec{n}(x)$, where $\vec{n}$ is a three-dimensional unit vector whose tip defines a point on the 2 sphere $S^2$. Suppose $\vec{n}$ tends to the same unit vector $\vec{n}(\infty)$ at large distance in all directions. We can identify the base manifold as $\mathbb{R}^2\cup \{\infty\}=S^2$: then we have: $$N=\frac{1}{4\pi}\int_{\mathbb{R}^2}\epsilon_{ijk}n^i\partial_xn^j\partial_yn^k dxdy\in \mathbb{Z}=\pi_2(S^2)$$ My question is:
Is it possible to write down similar simple expression for all homotopy group? For example, $\pi_4(S^3)=\mathbb{Z}_2$ and Hopf map $\pi_3(S^2)=\mathbb{Z}$ ?