How to pair up event participants?
You can get somewhat close to your desired functionality by putting the contacts in a smart group and displaying them within the relationship report.


Use the advanced search to display all contacts within the event (by adjusting the search fields within ("Events").
Add all the contacts to a new smart group.
Create a new report to show relationships (Administer > CiviReport > Create new report from template > Relationship report).
In "Filters", set "Group" to your smart group.
Adjust other report settings as necessary.


But I suspect the end result will still not be 100% satisfying. Groups of three might not display well enough for you.