For which orthogonal matrices does the matrix exponential converge?

I know that online sources such as Wikipedia and Wolfram just state without any proof or extended discussions that the matrix exponential is well-defined and converges for any square matrix.

$\quad$ Every matrix has an element of maximal size. $($Obviously, if anything can cause

divergence, it's that one$).~$ Let its absolute value be $M.~$ So let us construct a square

matrix S, whose every single element is $M.~$ Then $S^k=\Big(n^{k-1}M^k\Big)_{n\times n}~,~$ and each

element of $A^k$ lies between $\pm~n^{k-1}M^k.~$ But $~e^S\approx\bigg(\dfrac{e^{nM}}n\bigg)_{n\times n}~,~$ so every element

of $e^A$ is definitely bounded. However, even in this case divergence could still theoret–

ically happen, if at least one such element $($not necessarily the same$)$ were to freely

oscillate inside a given range, without actually converging to any particular value

within that interval. But this is not possible, since each new term of the infinite series

decreases at an exponential rate, being trapped between $\pm~\dfrac{n^{k-1}M^k}{k!}.$