Strassen has shown that the bound is higher than n^2 using a tensor rank bound. [1] The exact exponent bound is usually called omega in the literature, and there are good estimates for the value of omega. It is also related to other quantities like Grothendieck constant [2].

[1] http://www.thi.informatik.uni-frankfurt.de/~jukna/Strassen-V...

[2] https://arxiv.org/pdf/1711.04427.pdf