Oh, okay. That makes sense.
So for a given matrix a, we can convert a linear index n to a Cartesian index using:
CartesianIndices(a)[n]
Inversely, we can convert a Cartesian index (i,j) to a linear index using:
LinearIndices(a)[2,2]
This is exactly what I needed. One remaining question - do the above methods allocate a new matrix every time? That would be problematic.