# Exponent of matrix

I got this problem:

Let $A \in M_n(\mathbb R)$ such that $A = -A^T$. Prove that $e^A$ is an orthogonal matrix.

I succeeded in showing that $e^{(A^T)}e^A=I$ but did not succeed in proving that $e^{(A^T)}=(e^A)^T$. Any suggestions? Thanks to helpers!

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Consider definition and formula : $e^A =\sum_{i=0}^\infty \frac{1}{i!}A^i$ and $(XY)^T = Y^TX^T$ –  Hee Kwon Lee Feb 11 at 10:58
I still have a problem. Could you explain more? –  Shlomi Feb 11 at 11:07
I will add details : $$[e^A]^T = \bigg[\sum_{i=0}^\infty \frac{1}{i!}A^i \bigg]^T = \sum_{i=0 }^\infty \frac{1}{i!} (A^i)^T=\sum_{i=0 }^\infty \frac{1}{i!} (A^T)^i =e^{A^T}$$