For a quadratic space $(V, q)$ over a field $F$ (of characteristic $\neq 2$), we can define is associated special orthogonal group $\mathrm{SO}(V, q)$ and the general spin group $\mathrm{GSpin}(V, q)$. When $(V, q)$ is regular, i.e. if the associated bilinear map $B_q: V \times V \to F$ is non-degenerate, then the latter can be defined via Clifford algebra $C(V, q)$, which becomes a $\mathrm{GL}_1$ cover of $\mathrm{SO}(V)$:
$$ 1 \to \mathrm{GL}_1 \to \mathrm{GSpin}(V) \xrightarrow{p} \mathrm{SO}(V) \to 1. $$
The surjectivity of $p$ follows from the Cartan-Dieudonné theorem which assumes non-degeneracy of $B_q$.
I wonder what is the right definition of $\mathrm{GSpin}(V)$ for non-regular (degenerate) quadratic space $V$. Let's consider the most extreme case when $q = 0$, i.e. $V$ is totally isotropic. Then $\mathrm{SO}(V) = \mathrm{SL}(V)$. Clifford algebra still can be defined for degenerate spaces, and we have $C(V, 0) = \Lambda(V)$ (exterior algebra of $V$). We can still define $\mathrm{GSpin}(V)$ as regular cases:
$$ \mathrm{GSpin}(V):=\{ g \in C^+(V): \exists g^{-1}, gVg^{-1} = V\} $$
where $C^+(V)$ is the even Clifford algebra, and I guess that $C^+(V) = \Lambda^+(V)$ and $\mathrm{GSpin}(V) = \mathrm{GL}(\Lambda^+(V))$ for totally isotropic $V$. However, now I'm not sure if the map $p: \mathrm{GSpin}(V) \to \mathrm{SO}(V), g \mapsto (v \mapsto gvg^{-1})$, is still surjective and also have the same short exact sequence as above. More generally, I wonder if we have any theory and classification of low-rank general spin groups for non-regular quadratic spaces. Thanks in advance.