To put on an open-cell wetsuit, you have to use a lot of water (shower or bathe in it), and out in the field it will therefore be much better to have a thermos with 1-2 liters of warm water with added application agent or skin-friendly soap (e.g. intimate soap without perfume).
If you often do freediving, it is important to avoid shampoo/hand soap etc., otherwise you may eventually develop allergic reactions to these soaps. This may be due to perfume or other chemicals that have been added.
After the dive is over, you take off the suit and immediately rinse it in the sea. All equipment is then rinsed in fresh water (most importantly with suit, socks and gloves). Do not dry it in sunlight, as the UV light can break down the neoprene causing it to darken.
The suit is first dried with the wrong side out, to avoid bacterial growth. It is then dried with the outer side out. Should you be unlucky and get a bad smell in the suit, it can be washed in detergent and possibly deodoriser with the wrong side out.