TITLE: What linear transformations preserve these conditions?
QUESTION [5 upvotes]: Main Question
Let's define $\Gamma(n)$ as the set of real antisymmetric matrices of size $n$ ($n$ is an even Integer), fulfilling:
$$ \forall \gamma\in \Gamma(n) \Rightarrow \gamma^2=-\mathbb I_n$$ where $\mathbb I_n$ is the identity matrix of size $n$. What is a nice representation of the set of all linear operators that keep matrices inside $\Gamma(n)$?
$$\gamma_{i,j}\to \gamma'_{i,j} = \sum_{i',j'=1}^n \gamma_{i',j'}\beta_{i,i'}\beta_{j,j'}$$ where $\beta \in \mathbb C^{n^2}$.
For instance I know that if $\gamma\to \gamma'=U\gamma U^\dagger$, where $U$ is a unitary matrix then clearly $$\gamma'^2=U\gamma U^\dagger U\gamma U^\dagger=U\gamma^2 U^\dagger=-U\mathbb I_n U^\dagger=-\mathbb I_n$$ but I don't know what unitary matrices preserve antisymmetric-ness of a matrix. 

Background
The background is heavily related to physics so I thought I should separate it from the main question. The $\gamma$ matrices that I'm dealing with are called Covariance Matrices(CMs); and the case of this problem they are defined as:
$$\gamma_{l,m}=\frac{i}{2}\text{tr}(\rho[c_l,c_m])$$ where $\rho$ is the density matrix of a pure Gaussian state, $c$'s are Majorana mode operators fulfilling $\{c_l,c_m\}=2\delta_{l,m}$ and $[\_,\_]$($\{\_,\_\}$) is a/an (anti-)commutator. 
Also, the condition $\gamma^2=-\mathbb I_n$ is satisfied iff the state $\rho$ is pure[1].

REPLY [2 votes]: After a series of exchanges in the comment section, the OP has clarified that the true question is as follows:

Let $n$ be an even number and $\Gamma(n)\subset M_n(\mathbb R)$ be the set of all $n\times n$ real antisymmetric square roots of $-I_n$. Find all (complex) unitary matrices $\beta$ such that the mapping $\gamma\mapsto\beta\gamma\beta^\top$ (note: not $\gamma\mapsto\beta\gamma\beta^\dagger$ albeit $\beta$ may be complex) preserves $\Gamma(n)$.

We will prove the followings.

When $n\ge4$, the required matrices are given by those $\beta$s such that $\beta$ or $i\beta$ is a real orthogonal matrix.
When $n=2$, the required matrices are given by those $\beta$s such that $\beta$ or $i\beta$ is of the form
$$
\pmatrix{z\,\cos t&-\bar{w}\,\sin t\\ w\,\sin t&\bar{z}\,\cos t},
$$
where $t$ is an arbitrary real number and $z,w$ are arbitrary complex numbers of unit moduli.


Proof for $n\ge4$. Let $\mathcal K$ denotes the real linear space of all real antisymmetric matrices. Define $h=\beta^\top\beta$. Then $h$ is unitary and the condition $(\beta\gamma\beta^\top )^2 = -I$ implies that $h\gamma = \gamma h^{-1} = \gamma \bar{h}^\top$. Since $\mathcal K$ is the real linear span of $\Gamma(n)$, it follows that
$$
hk = k\bar{h}^\top\tag{1}
$$
for every $k$ in $\mathcal K$. Denote by $E_{ij}$ the matrix with a $1$ at the $(i,j)$-th position and zeros elsewhere. By considering all $k$s of the form $E_{ij}-E_{ji}$ and compare both sides of $(1)$ elementwise, it is easy to see that $h$ is a real multiple of the identity matrix (that $n>2$ is essential here). Yet, $h$ is unitary. Therefore $h=\beta^\top \beta=\pm I$.
If $\beta^\top \beta=I$, since $\beta$ is also unitary, we have $\beta^\top \beta=I=\beta^\dagger \beta$. Hence $\beta^\top=\beta^\dagger$, i.e. $\beta$ is real. Therefore $\beta$ is real orthogonal.
If $\beta^\top \beta=-I$, then $(i\beta)^\top(i\beta)=I$ and $i\beta$ is also unitary. So, the same argument shows that $i\beta$ is real orthogonal.
Proof for $n=2$. 
The set $\Gamma(2)$ has only two elements, namely, $g=\pmatrix{0&-1\\ 1&0}$ and $-g$. If the mapping $f:\gamma\mapsto\beta\gamma\beta^\top$ preserves $\Gamma(n)$, either $\beta g\beta^\top=g$ (and $f$ is the identity map) or $\beta g\beta^\top=-g$ (and $-f$ is the identity map). In the latter case, we have $(i\beta) g(i\beta)^\top=g$ and so it suffices to consider the former case only. Since $\beta$ is unitary, the condition $\beta g\beta^\top=g$ is equivalent to $\beta g=g\bar{\beta}$. By comparing elementwise both sides of this equality and also both sides of $\beta\beta^\dagger = I$, the result follows.