The quadratic function $f:\mathbb R^4\to\mathbb R^3$ $$f(a,b,c,d)=\begin{bmatrix} 2(ac + bd)&2(ad - bc)&a^2 + b^2 - c^2 - d^2\end{bmatrix}$$ surjectively maps the sphere $S^3$ to the sphere $S^2$. Interestingly, if one identifies $\mathbb R^4$ with $\mathbb C^2$, we can express $f$ as $$f(v)=\begin{bmatrix}v^*Xv&v^*Yv&v^*Zv\end{bmatrix}$$ where $X=\begin{bmatrix}&1\\1\end{bmatrix}$, $Y=\begin{bmatrix}&-i\\i\end{bmatrix}$, $Z=\begin{bmatrix}1\\&-1\end{bmatrix}$ are the Pauli matrices.
Can this phenomenon be generalized? In particular, for which $n$ does there exist a surjective quadratic function from $S^{2n-1}$ to $S^{2n-2}$?
