There's a really easy and mess free way, but I'm not sure I can describe it. I'll try . . .
Basically aim to chop your mango into three with the skin on - like in the image above, but done by knife. It's easy if you stand the mango upright and work out where the stone is. Then just slide the knife down along the edge of the stone, so that you have 3 sections - the middle one has the stone in.
Take one of the stone free pieces and hold it, skin down in the palm of your hand. Use a sharpish knife and slice up and down and side to side, so that you make a grid in it, reaching to, but not pressing hard enough to cut through, the skin (or through your hand behind it). The cubes you are cutting should be about 1.5 cms across.
Then push in the middle of the skin and your convex shape becomes concave - you are then able to run the knife over the inside of the skin and neatly chop off the cubes.
Repeat for the other piece without a stone in it and then simply peel off the little bit of skin on the stone piece. That one is for you - you bite/suck the mango flesh off from around the stone - cook's treat.
You get loads of mango off this way and should get almost none on your hands, until you gorge yourself on the stone piece!
HTH Enjoy