The skin around a wound needs its own care, separate from the wound itself. The advice pulls in two directions that are worth holding together: keep it clean, and keep it dry.
Cleaning it. With an open surgical wound, you may use a gauze pad or soft cloth to clean the skin around your wound. Soak the gauze or cloth in a normal saline solution or mild soapy water, and gently dab or wipe the skin with it. Try to remove all drainage and any dried blood or other matter that may have built up on the skin.
Dabbing and wiping, not scrubbing. With a closed surgical wound, do not rub or scrub the wound. Drying is the same. Gently pat dry the area around the wound with a clean towel, and do not rub it dry.
Keeping it dry, and why. With both venous and ischemic leg ulcers, keep the dressing and the skin around it dry. Try not to get healthy tissue around the wound too wet, because this can soften the healthy tissue, causing the wound to get bigger. Softened skin around a wound is not a cosmetic problem; it is how a wound enlarges.
Keeping it in good condition. With a venous ulcer, protect the skin around the wound by keeping it clean and moisturized. With pressure sores, caring for healthy skin by keeping it clean and moisturized is one of the ways to avoid further injury. If you have diabetes, use lotion, petroleum jelly, lanolin, or oil on dry skin on your feet, with one exception: do not put lotion, oil, or cream between your toes unless instructed to do so by your provider.
And what not to put near it. Do not put any lotion, cream, or herbal remedies on or around your wound unless your provider has said it is OK, and do not use lotions, powders, cosmetics, or any other skin care products on the wound itself.
Put together: clean the surrounding skin gently, dry it properly, moisturize healthy skin away from the wound, and keep moisture off the skin the dressing sits on.