There does not seem to be any consensus among languages and libraries:

Wikipedia wrote:Row-major order is used in C/C++/Objective-C (for C-style arrays), Mathematica, PL/I, Pascal, Speakeasy, SAS, and C#/CLI/.Net.

Column-major order is used in Fortran, OpenGL and OpenGL ES, MATLAB, GNU Octave, S-Plus, R, Julia, Rasdaman, Scilab.

Row-major order is the default in NumPy (for Python).

There is also the BLAS approach where operations take flags to indicate which matrices are transposed.