The symmetric space $\operatorname{SO}(n,\mathbb C) / \operatorname{SO}(n)$

Question

I am looking for a reference for the symmetric space $\operatorname{SO}(n,\mathbb C) / \operatorname{SO}(n)$; I haven't been able to find any references on about it online. In particular, I would like to know a "geometric interpretation" of the space.

By geometric interpretation, I mean a smooth manifold $M$ equipped with an action of $\operatorname{SO}(n,\mathbb C)$ with isotropy $\operatorname{SO}(n)$, something similar to the following list:

• $\operatorname{SL}(n,\mathbb R)$ acts on the symmetric positive-definite matrices of determinant one, via $A \cdot B := ABA^\top$. The isotropy subgroup at the identity is $\operatorname{SO}(n)$.

• $\operatorname{Sp}(2n,\mathbb R)$ acts on the symmetric $n$ by $n$ complex matrices with positive-definite imaginary part via $$\begin{pmatrix} A & B \\ C & D \end{pmatrix} \cdot Z := (AZ + B)(CZ + D)^{-1}.$$ The isotropy subgroup at $i I_n$ is isomorphic to $U(n)$.

• $\operatorname{SO}(p,q)$ acts on the positive-definite $p$-dimensional subspaces in $\mathbb R^{p,q}$ by $A \cdot V := AV$. The isotropy subgroup at $\mathbb R^p \oplus \{0\}$ is $S(\operatorname{O}(p) \times \operatorname{O}(q))$.

Here are some questions would could potentially help me:

1. If $X$ is an $n$ by $n$ real symmetric matrix, then $iX$ is Hermitian with imaginary entries. Since $iX$ is Hermitian, we know that the matrix exponential $\exp(iX)$ is positive-definite and symmetric. What does the "imaginary entries" condition of $iX$ tell us about $\exp(iX)$?
2. The Lie group $\operatorname{SL}(n,\mathbb C)$ acts transitively on the set of $n$ by $n$ (complex) positive-definite Hermitian matrices with determinant one, denoted $ P_1(n,\mathbb C)$. The action is given by $A \cdot B := ABA^\dagger$. Of course, since $\operatorname{SO}(n,\mathbb C)$ is a Lie subgroup of $\operatorname{SL}(n,\mathbb C)$, $\operatorname{SO}(n,\mathbb C)$ acts on $P_1(n,\mathbb C)$ via the same formula. The isotropy subgroup at $I_n \in P_1(n,\mathbb C)$ of this new action is clearly $\operatorname{SO}(n)$. What are the orbits of this new action?