First, I regard antibiotic ointment and bandages absolutely essential. I easily cut myself, being an artist and fixer-upper, but a total klutz. If I have a sharp knife, good chance I'll cut myself. I even get paper cuts. Minor or major cuts get washed with hot water and soap, then rubbing alcohol, then hydrogen peroxide, then covered with antibiotic ointment and bandage* til the wound is clearly healing. Especially vital if the skin is open, so it closes and heals correctly, which can take several days (I tend to cut deeply at times).
I grew up using antibiotic ointment, mostly Neosporin, but sometimes store brands. I read reviews of various products here, with short expiration dates. I figure if I spend half as much as at local drug store chain and get a small .5 oz tube, it balances spending twice as much which lasts longer. I'm ok with spending less at a time.
.5 oz usually lasts far longer than one year of my usage, even with my many, usual knife accidents. But due to short expiration, will be limited usage. BUT for the price, I am not complaining. Less than $4 for a year of healing against infections is ok with me.
*PS my late mother was a registered nurse. I worked with her at a major manufacturing plant. I watched her as she dealt with many cuts, plus of course, years of her taking care of my cuts as a child.