It is known that an entire function that is nowhere zero must be the exponential of another entire function.

Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire function, none of whose eigenvalues is zero anywhere (save at complex infinity, trivially), is it true that it must be the exponential of another matrix-valued entire function?

I need (not in mathematical sense) this to hold, because (in particular) it would imply that a suitable pointwise branch of "$\ln(e^{K}e^{L})$", $K$ and $L$ being real skew-symmetric matrices, would exist such that it is real analytic w.r.t. the (upper) elements of $K$ and $L$. (I tried to prove this weaker statement using the BCH-D formula with no apparent success)