  Let \(\mathcal{A}\) be a unital Banach algebra such that any Jordan
derivation from \(\mathcal{A}\) into any \(\mathcal{A}\)-bimodule
\(\mathcal{M}\) is a derivation. We prove that any 2-local derivation from the
algebra $M_n(\mathcal{A})$ into $M_n(\mathcal{M})$ $(n\geq 3)$ is a derivation.
We apply this result to show that any 2-local derivation on the algebra of
locally measurable operators affiliated with a von Neumann algebra without
direct abelian summands is a derivation.
