This may not be the 'official' right way to clean a wound, but....
I had to have an ingrown toenail removed on both sides of my big toe which left a 3 mm wide strip of exposed raw,bleeding skin along each side of the nail(I know...gross...) and as I was nursing, I didn't want to take antibiotics to prevent infection so, this 'old school' podiatrist had me do the following....
Soak in epsom salts 3x per day for 15 minutes, followed by putting betadine on the wound (I poured it over) and letting it sit for 5 minutes, then pour hydrogen peroxide over that, wrap up in gauze and tape it. Do this for a week.
Initially I thought the guy was a quack, but I tell ya - I didn't get an infection and it healed up quite nicely! I must admit I only did it twice per day also (was just too hard with a demanding 2 month old!)
So, likely there are many different ways to do it - this one may be more hard-core than you need, If it was me I would just pick whatever method seems to make the most sense to you and monitor the wound closely for redness, swelling, puss, etc. - anything to indicate it might be infected. Obviously, this is not medical advice....just an idea based on my previous experience.