Finding two skew symmetric matrices with same exponential value

Question

I've have the following problem.

I need to come with an example for two skew symmetric matrix $A,B$ such that $\exp(2\pi A)$ = $\exp(2\pi B)$
I tried matrix $2x2$ but I guess this is useless to just try, I probably need to construct the matrix by some thm.
Now, Only I know is that the function $F$ defined by $F(A) = \exp(A)$ has the next properties:

1. derivative near 0 is the identity matrix
2. $F$ itself is diffeomorphism of neighborhood contains 0 to neighborhood of $I_n$

How can I proceed from here?

Answer 1

Hint: If we let $J = \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}$, then $\exp(\theta J) = \begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos\theta\end{bmatrix}$.

Can you find two numbers $\theta_1 \neq \theta_2$ such that $\exp(\theta_1 J) = \exp(\theta_2 J)$?