Have you read "matrix multiplication by arithmetic progressions"? This paper currently holds the record for the lowest exponent for n*n matrix multiplication. The authors demonstrate the existence of, but do not describe, an algorithm and I doubt anybody has ever implemented it.
The limitations of my knowledge of bilinear forms makes it impossible for me not to take the result claimed on faith and the absence of any dispute about the result. I suspect most people have neither the time nor a large enough enough dense matrix to make an implementation worthwhile.