Let $M_n subseteq SO(2n)$ be the set of real $2n times 2n$ matrices $J$ satisfying $J + J^{T} = 0$ and $J J^T = I$. Equivalently, these are the linear transformations such that, for all $x in mathbb{R}^{2n}$, we have $langle Jx, Jx rangle = langle x, x rangle$ and $langle Jx, x rangle = 0$. They can also be viewed as the linear complex structures on $mathbb{R}^{2n}$ which preserve the inner product.

I’d like to understand $M_n$ better as a topological space, namely an $(n^2-n)$-manifold.

$M_1$ is just a discrete space consisting of two matrices: the anticlockwise and clockwise rotations by $pi/2$.

For $n geq 2$, we can see that $M_n$ is an $M_{n-1}$-bundle over $S^{2n-2}$. Specifically, given an arbitrary unit vector $x$, the image $y := Jx$ must lie in the intersection $S^{2n-2}$ of the orthogonal complement of $x$ with the unit sphere $S^{2n-1}$. Then the orthogonal complement of the space spanned by $x$ and $y$ is isomorphic to $mathbb{R}^{2n-2}$, and the restriction of $J$ to this space can be any element of $M_{n-1}$.

Since the even-dimensional spheres are all simply-connected, it follows (by induction) that $M_n$ has two connected components for all $n in mathbb{N}$, each of which is simply-connected. For instance, $M_2$ is the union of two disjoint 2-spheres: the left- and right-isoclinic rotations by $pi/2$. The two connected components of $M_n$ are two conjugacy classes in $SO(2n)$; they are interchanged by conjugating with an arbitrary reflection in $O(2n)$.

Is (each connected component of) $M_n$ homeomorphic to a known well-studied space? They’re each an:

$S^2$-bundle over an $S^4$-bundle over $dots$ an $S^{2n-4}$-bundle over $S^{2n-2}$

but that’s not really very much information; can we say anything more specific about their topology?