Matrix square roots of -I Since we can see matrices as generalizations of complex numbers, I asked myself if there is a way to classify those matrices which are the "Basis" for the complex part. That is, I would like to identify the set of $n\times n$ real valued matrices $M$ whose square $M^2$ is equal to $-I$, where $I$ is the $n\times n$-identity matrix. Is this set already classified?    
The matrix $J = \left( \begin{smallmatrix}0 & -1\\1 & 0 \end{smallmatrix} \right)$ satisfies $J^{2} = -I$ in the 2-dimensional case. 
EDIT:
Thanks to your comments and answers I have the following observation (if its wrong, please tell me): 
Suppose that $M$ satisfies that $M^t=-M$, then we can observe for two vectors $x,y\in\mathbb{R}^{2n}$ that

$y^tAx=(Ax)^{t}y=x^{t}A^ty=-x^tAy$

My interpretation is that if the angle between $x$ and $Ay$ is $\alpha$ then $\pi+\alpha$ is the angle between $Ax$ and $y$, since ${\displaystyle \cos \;x=-\cos(x+\pi )}$ (here we suppose that the length of the vectors are not relevant). If now $y=x$ and then we can see that $Ax$ is orthogonal to $x$. 
Is it true that $A^2$ will be a rotation of 180 degrees, right? Because of vector length we will have something like $A^2=-CI$, where $C$ is just a constant. Is that true?

You might be interested in complex structures on real vector spaces.
The basic idea is this: suppose I have a complex vector space $V$ of dimension $n$. If I forget how to multiply by $i$, and only use scalars that are real, then $V$ is also a real vector space of dimension $2n$ (let's call this $V_{\mathbb{R}}$, to make it clear I've forgotten anything complex). An interesting question is: what exactly did I forget when I forgot how to multiply by $i$? Can I remember it for later, and get back my original complex vector space?
On the complex vector space $V$, there is a linear map $J: V \to V$ which is multiplication by $i$, defined by $Jv = iv$. In fact, this map is $\mathbb{R}$-linear, and so gives a linear map $J: V_\mathbb{R} \to V_\mathbb{R}$. We can check that $J^2v = i^2v = -v$, and so $J^2 = -\mathrm{id}_V$. This map $J$ is exactly what we forgot when we forgot how to multiply by $i$: if I have a vector $v \in V_{\mathbb{R}}$ and want to scale it by $(a + bi)$, then I use $(a \,\mathrm{id}_V + bJ)v$.
Your question is essentially going the opposite way: starting with a $2n$-dimensional real vector space $W$, what are the kind of maps $J: W \to W$ such that $(W, J)$ becomes a complex vector space with action $(a + bi)w = (a \, \mathrm{id}_W + bJ)w$.
