One can generalize the notion of $n$-Lie algebra (in the sense of Fillipov) and define ''weak $n$-Lie algebra'' to be an anticommutative $n$-ary algebra $(A,[\cdot ,\ldots ,\cdot ])$ such that the $% (n-1)$-ary bracket $[\cdot ,\ldots ,\cdot ]_a =[\cdot ,\ldots ,\cdot ,a]$ is an $(n-1)$-Lie bracket on $A$ for all $a$ in $A$. It is well known that every $n$-Lie algebra is weak $n$-Lie algebra. Under some additional assumptions these notions coincide. We show that it is not the case in general. By means of representation theory of symmetric groups a full description of $n$-bracket multilinear identities of degree $2$ that can be satisfied by an anticommutative $n$-ary algebra is obtained. This is a solution to the conjectures proposed by M. Bremner. These methods allow us to prove that the dual representation of an $n$-Lie algebra is in fact a representation in the sense of Kasymov. We also consider the generalizations of $n$-Lie algebra proposed by A. Vinogradov, M. Vinogradov and Gautheron. Some correlation between these generalizations can be easily seen. We also describe the kernel of the expansion map.