An $n$-ary quasigroup $(Q,A)$ such that for some $i\in\{1,\dots,n\}$ the identity \[ A(A(x^n_1),A(x^{n-1}_2,x_1),\dots,A(x_n,X^{n-1}_1))=x_i \] holds is called an $i$-Weisner $n$-quasigroup ($i$-W-$n$-quasigroup). $i$-W-$n$-quasigroups represent a generalization of quasigroups satisfying Schröder law $(xy\cdot yx=x)$ and quasigroups satisfying Stein's third law $(xy\cdot yx=y)$. Properties of $i$-W-$n$-quasigroups which are satisfied for all i are determined. Necessary and sufficient conditions for an $n$-quasigroup to be an $i$-W-$n$-quasigroup are obtained. It is proved that every $i$-W-$n$-quasigroup of order $v$ defines an orthogonal set of $n$ $(n-1)$-quasigroups of order $v$. It is shown that some $i$-W-$n$-quasigroups are equivalent to orthogonal arrays. Conjugates of $i$-W-$n$-quasigroups are investigated, connections among these conjugates for different values of $n$, $i$ are established. The existence of several classes of $i$-W-$n$-quasigroups is proved.