There is the famous Strassen matrix multiplication method (see here for further information).
In essence it boils down to the fact, that we can multiply 2x2 matrices with each other and don't have to use 8 multiplications, but $7$ instead.
When we would work with $3\times3$ or $4\times4$ etc. matrices instead, is there a way that we have to use significantly fewer multiplications than $n^3$.