Well, your existing board looks like it's a rectilinear projection of the cube centred at one of its corners with the projection point at the opposite corner (Which makes it very similar to a stereographic projection). That can be extended to cover the entire cube except for the corner opposite the one in the centre, which would end up at infinity, and the 3 spaces sharing it would be stretched around the entire perimeter of the board. They would also be extending to infinity, and the squares near them would be huge compared to those int he middle. An equidistant azimuthal projection would mitigate that though it would be more complex to figure out as you would have to project everything from the cube onto the cube's circumsphere first.