  Let $L$ be a Lie algebra over a field of characteristic different from $2$.
If $L$ is perfect and centerless, then every skew-symmetric biderivation
$\delta:L\times L\to L$ is of the form $\delta(x,y)=\gamma([x,y])$ for all
$x,y\in L$, where $\gamma\in{\rm Cent}(L)$, the centroid of $L$. Under a milder
assumption that $[c,[L,L]]=\{0\}$ implies $c=0$, every commuting linear map
from $L$ to $L$ lies in ${\rm Cent}(L)$. These two results are special cases of
our main theorems which concern biderivations and commuting linear maps having
their ranges in an $L$-module. We provide a variety of examples, some of them
showing the necessity of our assumptions and some of them showing that our
results cover several results from the literature.
