How to find all skew symmetric matrices of $O(3)$? $O(3)$ is the group of all $3\times 3$ orthogonal matrices, skew symmetric means $A^t=-A$. A hint or general approach is appreciated, am not necessarily looking for complete solution. 

$A$ orthogonal implies $$AA^t=I$$
and using the skew-simmetry substituting $A^t$ $$A^2=-I$$
Now consider an eigenvalue $\lambda$ of A. Then $\lambda ^2$ is an eigenvalue of $A^2$ ($A$ must have at least one real eigenvalue, because the characteristic polynomial of $A$ is of degree three). But $-I$ has only negative eigenvalues. Then there is no such $A$.
