Should you be concerned if you notice a small bruise with a white circular middle that doesn't itch or bleed and could this be some form of skin cancer?

Anything is possible and you should see a doctor immediately to have it diagnosed. They will take a tissue sample to see if it's cancerous or not. Even if it is skin cancer is one of the cancers that is most curable. My friend's husband just had a minor operation on his hand (skin cancer) and it didn't hurt and all is well and should remain well, but when he keeps going in for his physicals each year his doctor will make more of an effort to check out his body for possible skin cancer. You could have Excema or Psoriasis, still, it's important to get it checked out.