Designing flyback converter - wire selection
You often use multiple strands of wire to reduce the effect of skin depth. The skin depth itself will stay the same.

The way this is normally done is to use 'Litz' wire, which is made up from multiple strands, sometimes more than 100 individual strands. You can also make your own by twisting together a number of strands, though it probably won't be as tidy as commercial Litz.

The wire used has to have self-fluxing solderable enamel. Although most wire has this these days, you may pick up some very old stock that doesn't solder well. This is essential as it's impractical to terminate all strands manually. With self-fluxing, you heat the end of the wire together with some multicore solder, and the enamel burns back and allows the wire to tin.

Litz wire is only beneficial over a limited range of frequencies. At low frequencies, audio and below, skin depth is rarely a problem for practical wire thickness. At high frequencies, more than a few MHz, the skin depth is so small that Litz can't be made fine enough to benefit, given that its construction includes much insulation and air, and it's best to go back to a single wire. Within that frequency range, check skin depth tables for the wire diameter you want to use, to see whether it's worth using.

All Litz wire does is to reduce the copper loss. This is only one component of the system loss, the other important contributors being ferrite core loss, and semiconductor conduction and switching losses. It will make your transformer run slightly cooler. Whether that's needed, or a 'nice to have', is down to the detail of your system. Unless you are shooting for the ultimate in efficiency, it's usually possible to use single core. 

A slightly different application where you sometimes must use Litz is when winding inductors intended for LC filters. The loss impacts the acheivable Q, a much more fundamental spec point than flyback efficiency.