  In the present paper we prove that every 2-local inner derivation on the Lie
ring of skew-adjoint matrices over a commutative $*$-ring is an inner
derivation. We also apply our technique to various Lie algebras of
infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that
every 2-local spatial derivation on such algebras is a spatial derivation. We
also show that every local spatial derivation on the above Lie algebras is a
derivation.
