Do you think it's infected?

On the morning of January 1st I accidentally kicked/bumped my foot on the faucet in my bathtub. I stubbed my 2nd toe's toenail on it. It hurt for a moment like I stubbed my toe but the pain went away a few seconds later.

That night I look at my toe and there was blood everywhere, the toe was cold to the touch and the blood looked gooey. I didn't even notice all day that it was like that.

A little bit later the nail started to throb and when I pressed on it pus oozed out which freaked me out so I quickly cleaned it with antiseptic whipes and covered the wound in a few bandaids.

I cleaned the wound and changed the bandages for the past 2 days. Last night it started throbbing again and now the toe is inflamed and my toenail hurts when I step down on that foot.

That whole toe was cold to the touch when I first discovered the blood, but it is just normal body temp now. not hot, or unusually warm.

I would also like to point out that I have had the opportunity to not wear socks or shoes for the past few days which means my toe was not incubating in my sweaty shoes all day.

I hope it's not infected but I've never had an actual infection before so I need some advice please.

it is now jan 3th btw to add a time frame.