in this paper I found some explicit generators of homotopy groups of unitary groups, for example
$\pi_3[SU_2]$:
$\begin{bmatrix}z_1\\z_2\end{bmatrix}$$\rightarrow $$\begin{bmatrix}z_1 &-\bar{z_2}\\z_2 &z_1\end{bmatrix}$
and
$\pi_5[SU_3]$:
$\begin{bmatrix}z_1\\z_2\\z_3\end{bmatrix}$$\rightarrow $$\begin{bmatrix}z_1+\bar{z}_3z_2 & -\bar{z}_3^2 & -\bar{z}_2+\bar{z}_3\bar{z}_1\\z^2_2 & z_1-z_2\bar{z}_3 & z_3+z_2\bar{z}_1 \\ -z_3+\bar{z}_1z_2 & -\bar{z}_2-\bar{z}_1\bar{z}_3 & \bar{z}^2_1\end{bmatrix}$.
I am in search of explicit 2x2 unitary matrices of this form for $\pi_4[SU_2]$ and $\pi_5[SU_2]$, but haven't been able to find them (maybe they don't exist). If anyone knows of a paper containing them or an easy method of constructing them, I would greatly appreciate it.
Many thanks,
Pascal