If eight people participate in a Secret Santa gift exchange, what is the probability that the gift-giving will form a loop?

A Secret Santa gift exchange is when a group of people are randomly assigned another person to whom they give a gift.

A group of eight people have their names placed in a hat. Each person draws a random name from the hat. If a person accidentally draw their own name, they place it back into the hat and draw again. In end, each person will give a gift to one other person, and each person will receive a gift from one other person.

What is the chance that the gift-giving forms one large loop?

For example:
1 gives to 4
4 gives to 6
6 gives to 3
3 gives to 8
8 gives to 2
2 gives to 7
7 gives to 5
5 gives to 1

Also, how is this probability influenced by the number of people in the group?