No, not if you mean standard matrix multiplication. The matrix product of two $n\times 1$ or $1\times n$ matrices does not exist. Even if you allow taking the transpose, the product of a $1\times n$ matrix and a $n\times 1$ matrix is $1\times 1$ while the product of a $n\times 1$ matrix and a $1\times n$ matrix is $n\times n$. None of these are vectors, so this can't be what you are looking for.
However, as Andreas pointed out, there is a name for the operation you are talking about, the Hadamard product, and it generalizes to all matrices.