Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in G, t\in {\mathbb R}$.

Question. What are vector subspaces in $C$? For instance, are there 3-dimensional linear subspaces? (One can ask the same question for cones over other linear Lie groups $G$.)

Update: @Dmitri noticed that in dimensions divisible by $4$ and $8$ there are 4-dimensional and 8-dimensional linear subspaces coming from quaternions and octonions. Thus, the question is: What are other linear subspaces, in particular, are there linear subspaces (which are not lines) for odd $n$? Are there linear subspaces which are not contained in cones over subgroups?

Trivial examples of subspaces in $C(G)$ are planes, obtained as cones over $O(2)$. Diagonal embedding of $O(2)$ to $O(2n)$ (and its conjugates) yield examples of planes in $C(O(2n))$. Are there other examples of linear subspaces (of dimension $>1$) in $C(O(n))$? My motivation for the question comes from hyperbolic geometry (where the question is about affine subspaces of $O(n,1)$), but I find the linear algebra question about the orthogonal group intriguing, because it is so basic. The question is somewhat similar to the one in Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits

Alternatively, and algebraic geometers like it better, one can work over ${\mathbb C}$ and projectivize everything. Thus, I am asking about linear subspaces of the variety $X=PO(n)\subset {\mathbb P}^{N}$, $N=n^2-1$. In other words, the question is about Fano varieties $F_k(X)$ for various $k$. Algebraic geometers studied Fano varieties $F_k(X)$ of projective varieties since 19th century, so, maybe somebody looked at the case of homogeneous $X$, in particular, projective varieties $X$ which are algebraic subgroups. Unfortunately, google search did not yield anything useful.