Let $M=I+A\in \mathbb{R}^{n\times n}$ for a skew-symmetric matrix $A$ with $\|A\|<1$ in the spectral norm. Using the $LU$-decomposition of $M$, it is easy to construct a solution $L,U\in \mathbb{R}^{n\times n}$ with $L$ strictly lower- and $U$ upper triangular of $$ LM+MU=I. $$ Numerical tests suggest a bound of the form $\|L\|+\|U\|\leq C\log(n)$. Is that true? Moreover, these tests show that $\|L\|+\|U\|$ behaves much nicer than the $LU$-decomposition of $M$, hence a proof relying on the decomposition might not be possible.