Use a very clean cloth, do not risk introducing any dust or other contamination onto the cloth in case you scratch the screen. It sounds silly, but my wife will often clean stuff with a cloth she has not checked the "face" of.
I had some finger marks on mine and I didn't even bother with detergent. I used a soft dry paper towel and just rubbed very slowly in the area. Either way, patience is the key. I avoided detergent because of how hard the water is round here (it has tattoos and body piercings) and I was worried about leaving a stain from that.