What should I do to cover up my cut?

I just cut deeper than I ever have before. The skin is white and it’s bleeding a lot (I think it’s what people call styro?) I don’t have access to bandaids or any medical supplies right now. I already cleaned my arms with soap and water and dried them. Is it fine to just wear long sleeves? Will it heal fine and not get infected? If I should get some medical supplies, the soonest I can is probably around 10 hours from now.