I'm pretty sure that you have to organize the event yourselves as I don't think chess.com has an elimination-style tournament like you're describing. You could set up matches through your club's chat and use a bracket online or on paper to record the results.
How to set up a tournament for my club players
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I am pretty new to chess and have just set up a club so that I can organize a fun chess tournament amongst my work collegues. Let's say 12 people sign up for it, what I would like to do is have a "Wimbledon tennis" style tournament where people progress through rounds and into semis and the final. I would like people to play 10 minute matches of (say) 2 or 3 games on-line at time(s) of their mutual convenience and ideally have chess.com send out match invites/requests and ideally show a tree diagram of how everyone is doing. Maybe I hope for too much! The tournament stuff I have found seems to need to happen at a set time. Any ideas? Thanks!!