  We introduce the notion of weak-2-local derivation (respectively,
$^*$-derivation) on a C$^*$-algebra $A$ as a (non-necessarily linear) map
$\Delta : A\to A$ satisfying that for every $a,b\in A$ and $\phi\in A^*$ there
exists a derivation (respectively, a $^*$-derivation) $D_{a,b,\phi}: A\to A$,
depending on $a$, $b$ and $\phi$, such that $\phi \Delta (a) = \phi
D_{a,b,\phi} (a)$ and $\phi \Delta (b) = \phi D_{a,b,\phi} (b)$. We prove that
every weak-2-local $^*$-derivation on $M_n$ is a linear derivation. We also
show that the same conclusion remains true for weak-2-local $^*$-derivations on
finite dimensional C$^*$-algebras.
