ij=k means that the product of two of the three unit vectors must equal the third, which is a fundamental, non-arbitrary property, and without loss of generality the signs can be chosen so that ij=k (rather than ij=-k).
Adding ii=-1,jj=-1,kk=-1 it's easy to deduce ijk=-1, j=ki, and i=jk, and ij=-ji, ik=-ki, jk=-kj.
Having to remember, out of the six possible permutations, that one of the three correct formulas is the one with the symbols in alphabetical order is much better than remembering the "right hand rule" (or was it "left hand rule"?).
Having to remember, out of the six possible permutations, that one of the three correct formulas is the one with the symbols in alphabetical order is much better than remembering the "right hand rule" (or was it "left hand rule"?).