Existence of an $n\times n$ real matrix $A$ such that $A^2=-I$.

We can use the following characterization of diagonalizable:

A matrix or linear map is diagonalizable over the field F if and only if its minimal polynomial is a product of distinct linear factors over F.

$A^2+I=0$ implies that $A$ satisfies the polynomial $x^2+1$ which is the minimal polynomial since it is irreducibe over the reals. This minimal polynomial is not a product of distinct linear factors over $\mathbb{R}$, thus $A$ is not diagonalizable.

Invertible, skew-symmetric, orthogonal can be all ruled out with the example $A=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$.

Symmetric is false, by stity's comment below, since symmetric matrices are diagonalizable.

Alternatively, you can observe that it is false for $n=2$ by the computation: $\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}a&c\\b&d\end{pmatrix}=\begin{pmatrix}a^2+b^2&\dots\\\dots&\dots\end{pmatrix}=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$ is not possible over the reals.

I experimented with rotation matrices and powers and found the following solution for $\small n=4$ where $\small w = \sqrt{1/3}$ :

     0   w  -w  w
A=  -w   0   w  w
     w  -w   0  w
    -w  -w  -w  0

Then

      -3*w^2       .       .       .
           .  -3*w^2       .       .
  A^2=     .       .  -3*w^2       .  = - I
           .       .       .  -3*w^2

And also,

• $\small A$ is invertible
• $\small A^{-1} = A^T$ (or $\small A \cdot A^T=I $ ) which is a condition for orthogonal/orthonormal matrices.
• Also $\small A^T = - A$ which means, $\small A$ is skew-symmetric.
• Using complex numbers, a diagonalization is possible and gives the diagonal $\small D = [1,1,-1,-1] \cdot \sqrt{-3} $