There are many ways to go about remedying a mistake at work, but the most effective way — by far — is to hold yourself accountable for it. When you make a mistake take 5 minutes to center yourself on a way to approach it:
Staying accountable means that there is no time wasted being badgered by angry coworkers and switches the focus from your mistake to the next move forward. By owning your choices (and mistakes) you are able to save time, learn and ensure success going forward. No Nonsense. No Excuses.