Alternating symmetric $(AS)$ $n$-quasigroups are defined and considered. An $n$-quasigroup $(Q,f)$ is caled an AS-n-quasigroup iff $f(x_1,\dots,x_n)=x_{n+1}\Leftrightarrow f(x{\sigma_1},\dots x_{\sigma_n})=x_{\sigma(n + 1)}$ for every even permutation $\sigma$ of the set $\{1,\dots,n+1\}$. AS-n-quasigroups represent a generalization of semisymmetric quasigroups. Several equivalent definitions of an AS-n-quasigroup are given and it is proved that every AS-n-quasigroup, $n>3$, defines a family of totally symmetric $(n-2)$-quasigroups. Some properties of $(i,j)$-associative AS-n-quasigroups are determined and full characterization of AS-n-groups is given. Autotopisms and isotopism of AS-n-quasigroups are considered. Necessary and sufficient conditions for a principal isotope of an AS-n-quasigroup to be an AS-n-quasigroup are given.