If $V = \mathbb{R}^3$ with the Euclidean inner product $g$, and $S = \mathbb{C}^2$ is the corresponding space of spinors, then there is a quadratic map $h: S \to V^*$, which maps $\psi \in S$ to $h(\psi) \in V^*$, where
$$ h(\psi)(v) = i(v.\psi, \psi),$$
where the dot denotes Clifford multiplication, and $(-,-)$ denotes the "standard" hermitian inner product on $S$. Note that the right-hand side is real since Clifford multiplying by $v$ is skew-hermitian. This map is basically the Hopf map.
A colleague of mine confirmed that the definition of $h$ carries through to higher dimension, as I was suspecting. So in general, we assume that $V$ is a real vector space of dimension $n \in \mathbb{N}^*$ endowed with a symmetric positive-definite inner product $g$, and that $S$ is the corresponding space of spinors, which is a complex representation of $\operatorname{Spin}(n,\mathbb{R})$ of dimension $\lfloor \frac{n}{2} \rfloor$ (the "floor" of $n/2$). Then, we denote by $(-,-)$ an hermitian product on $S$ with respect to which $\operatorname{Spin}(n,\mathbb{R})$ acts unitarily, and with respect to which Clifford multiplication by a unit vector $v \in V$ is skew-hermitian. We then define the spinorial generalization $h$ of the Hopf map by the same formula (written in the first paragraph of this post).
It is easy to see that $h$ is $\operatorname{Spin}(n,\mathbb{R})$-equivariant, from which one may deduce that $h$ is onto.
Question: What is the fiber $h^{-1}(v)$ of $h$ over a point $v \in S^{n-1} \subset V^* \simeq \mathbb{R}^n$, with $v$ being thus of unit norm with respect to $g$?
Edit: upon asking prof. Bryant, he mentioned to me that my map is related to the squaring map. I could not find online a definition of the squaring map in general. It is a quadratic map from the space $S$ of spinors to the exterior algebra $\Lambda^*V^*$ which is further $\operatorname{Spin}(n,\mathbb{R})$ equivariant. I have an idea how to define such a map. Let $\alpha = v_1 \wedge \ldots \wedge v_k \in \Lambda^k V$ be a $k$-multivector. Denote by $Q: \Lambda^*V \to Cl(V,g)$ the "quantization" map. Define $h: S \to \Lambda^* V^*$ by
$$ h(\psi)(\alpha) = i^\frac{k(k+1)}{2} (Q(\alpha).\psi,\psi).$$
Then I believe $h$ has the required properties of the squaring map, but I am not sure if my definition agrees with that of the squaring map. Could someone please refer me to the definition of the squaring map or, better, possibly write it down?
Edit 2: I found the definition of the squaring map in the following article: https://arxiv.org/pdf/1709.02762.pdf in equation $12$. The spinorial generalization of the Hopf map $h$ in this post is essentially the $p = 1$ special case of formula $12$.