Here is how you play thanks to the website googolpower. Similar to the game rock, paper and scissors, students pair up to play a game where on the count of three they each randomly put out fingers from 0-5 (or 0-10 if they have more advanced skills.) Then it is a race to answer the math fact.
For example if child # 1 holds out five fingers and child #2 holds out three fingers, they would yell out the product 15. This game can also be played with younger students to reinforce adding skills. You can also play in teams or groups of three by having the third person answer the questions and taking turns.
Here are some more links to online multiplication games and practice:
Multiplication Idea Bank
Kids Games for Learning