Consider redundancy analysis
Redundancy analysis (RDA) is an extension of multiple linear regression (MLR) that accounts for multiple response variables as well as multiple explanatory variables. This direct gradient analysis technique attempts to effectively ordinate objects (the rows of your tables or matrices) on axes that are built to maximise their relationship to linear combinations of your explanatory variables. That is, the RDA axes and linear combinations of your explanatory variables should be maximally 'redundant'. The significance of this association, usually tested by permutation, can also be reported.
RDA is particularly useful where short gradients have been analysed, such as in a pilot study or in smaller experimental investigations.