Alternative scalar product for vectors

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Simple question. Is there a product between two vectors, $\mathbf{A}$ and $\mathbf{B}$ in which $$ \mathbf{AB}=A_iB_j$$ Where the subindex is a sum for all components of the vectors. Also is the definition $\mathbf{[A,B]}=\mathbf{AB}-\mathbf{BA}$ correctly expressed?