It's been a long time since I really looked at the phenomena, but I don't believe that current density is at unity until you get to that depth, then dropping to zero. I expect that it is a logarithmic decline as you go into the wire, starting from the first micron.
As such, you don't want to choose a wire whose diameter falls is anywhere close to the skin depth number. Far better is to be a fraction of that number. Then use multiple strands to carry the current.
This is a common practice in high frequency transformers and inductors, where "Litz wire" is often used. Google that and read the wikipedia article on the subject. For your frequency, you don't need to use the fine stranding (some as small as 40awg) often used in litz wire, but probably something in the 18 to 12 awg might be good.