A Skew-symmetric matrix  A satisfies the relation   $A^2+I=0$, when I is a unit matrix then A

A is skew-symmetric,    $A^T=-A$

$$\begin{gathered} A^2+I=0 \\ {A}^2=-I\Rightarrow AA=-I \\ A(-A)=I \\ A{A}^T=I \\ \Rightarrow A is orthogonal matrix \end{gathered}$$

$\left|A\right|=\left|A^T\right| and \left|KA\right|=\left|K^{n}A\right|$

$$\begin{gathered} A^T=-A \\ \left|A^T\right|={(-1)}^n\left|A\right| \\ \left|A\right|={(-1)}^n\left|A\right| \\ (1-{(-1)}^n)\left|A\right|=0 \\ \Rightarrow \left|A\right|=0or1-{(-1)}^n=0 \end{gathered}$$

n is even
But  
$A^2=0-I=-I$ 
 $\left|A^2\right|={(-1)}^n\left|I\right|\ne 0 \Rightarrow \left|A\right|\ne 0$
Hence, the only possibility is that A is of even order