Alternating symmetric $n$-quasigroups were introduced and described by Z. Stojaković in [9] es a generalization of semisymmetric (binary) quasigroups. In this note we generalize a part of the results obtained by Z. Stojaković and give some connections between alternating symmetric and totally symmetric $n$-quasigroups. Special elements in alternating symmetric $n$-quasigroups are described, too.