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: $A$ 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 $A$.
A: Out of curiosity, me too, I wondered what happens for the same condition (being antisymmetric and othogonal), this time in $\mathbb{R}^4$. The answer is interesting, the set is far from empty and depends on two angular parameters.
Even if it is not directly an answer but an extension of the question that has been asked, I place it here because the place devoted to "comments" is not large enough.
After some computations (using Mathematica), I have obtained:
$$\pm\begin{pmatrix}
0 & -\sin \beta & -\cos \beta \cos \alpha & -\cos \beta \sin \alpha \\
\sin \beta & 0 & \cos \beta \sin \alpha & -\cos \beta \cos \alpha \\
\cos \beta \cos \alpha & -\cos \beta \sin \alpha & 0 & \sin \beta \\
\cos \beta \sin \alpha & \cos \beta \cos \alpha & -\sin \beta & 0
\end{pmatrix}$$
When this matrix is decomposed into four  $2 \times 2$ blocks, these blocks have an interesting structure:
$$\pm \begin{pmatrix}
\sin \beta J & -cos \beta R_{-\alpha} \\
\cos \beta R_{\alpha} & -\sin \beta J 
\end{pmatrix}$$
where $R_{\alpha}$ is the rotation associated with angle $\alpha$, and $J= \begin{pmatrix}
0 & -1 \\
1 & 0 
\end{pmatrix}$ is the rotation associated with angle $+\dfrac{\pi}{2}.$
I would be very glad if somebody can bring some reflexion about this structure.
Knowing in particular that the exponential of any antisymmetric matrix is an orthogonal matrix, and that all othogonal matrices can be obtained in this way, one should take advantage of the upsaid block structure... 
