In northern Europe we do not need to use an antibiotic cream or ointment unless the wound is turning nasty.
In most cases all that should be done is to clean it with clean water, saline solution is even better. Cover with a plaster or bandage.
This on small, shallow cuts and abrasions. The deep ones, that might need suturing, need to be cleaned, bandaged, then seek professional help.
After a good cleaning, let the wound bleed for a bit. Think internal cleaning. Blood contains antibodies that will take care of bacteria.
The only wounds I would use an antibacterial cream/ointment on is wounds on the fingertips, around the nails and on the feet.
Finger and toe tips are sensitive if you get an infection, due to the anatomy they can go deeper quickly.
I personally prefer Neosporin. I carry that plus a 14 day course of Augmentin XR ( Amoxicillin plus Clavulanic acid, slow release) which is a very good broad spectrum antibiotic. I carry so I can take one tablet every 8 hours as opposed to the standard one tablet every 12 hours.
The idea is to reach an effective plasma concentration quickly, and have a high, bacteriecid concentration.
Betadine is an Iodine preparation, very good unless you are allergic or sensitive to Iodine.
As I live in the tropics where wounds are more sensitive, I use Neosporin if I have scratched or cut myself while outside.