Crooks is alleged to have opened fire while the former US president was addressing a crowd in Butler, Pennsylvania, leaving one audience member dead and two others badly hurt.
The 20-year-old was shot dead at the scene by a Secret Service sniper, officials said.
Thomas Crooks had not been carrying ID, so investigators used DNA to identify him, the FBI said.
He was from Bethel Park in Pennsylvania, about 70km (43 miles) from Butler, the site of the attempted assassination, and appears to have graduated in 2022 from Bethel Park High School, the school district confirmed in a statement.
State voter records show that Crooks was a registered Republican, according to US media.
He is also reported to have donated $15 to liberal campaign group ActBlue in 2021.
Law enforcement officials believe the weapon used to shoot at Donald Trump was purchased by Crook’s father, the Associated Press news agency reports.
Speaking on condition of anonymity, two officers told AP that Crooks’ father bought a weapon at least six months ago.
What was his motivation?
Having established Crook’s identity, police and agencies are investigating his motive.
“We do not currently have an identified motive,” said Kevin Rojek, FBI Pittsburgh special agent in charge, at a briefing on Saturday night.
The inquiry into what took place could last for months and investigators would work “tirelessly” to identify what Crooks’ motive was, Mr Rojek said.
Speaking to CNN, Crooks’ father, Matthew Crooks, said he was trying to figure out “what the hell is going on” but would “wait until I talk to law enforcement” before speaking about his son.
Police sealed off the road to the house where Crooks lived, CBS News reports.
A neighbour told CBS that officers evacuated her in the middle of the night with no warning.
Bethel Park Police said there was a bomb investigation surrounding Crooks’s home.
Did he hit anyone?
One person was killed and two others were injured in the shooting.
All three victims are adult men and were audience members, CBS News reports. Their names have not been released.
A GoFundMe page, organised by the Trump campaign’s national finance director Meredith O’Rourke, was set up in the hours after the attack with donations going to the families of the injured.
It has so far raised more than $340,000 (£267,000).
In a post to his Truth Social platform, Trump said he was “shot with a bullet that pierced the upper part of my right ear”.
“I knew immediately that something was wrong in that I heard a whizzing sound, shots, and immediately felt the bullet ripping through the skin,” Trump wrote. “Much bleeding took place, so I realised then what was happening.”
Blood was clearly visible on Trump’s ear and face as protection officers rushed him away.
Trump is “doing well” and is grateful to law enforcement officers, according to a statement published on the Republican National Committee (RNC) website.
How far was the suspected gunman from Donald Trump?
One witness told the BBC that he had seen a man – believed to be Crooks – with a rifle on the roof of a building before Trump was shot at.
BBC Verify analysed footage and confirmed that the gunman opened fire from on top of a flat warehouse building less than 200m from the former president.
Video footage obtained by TMZ shows the moment the shooting began.
The assailant opened fire with “an AR-style rifle”, the BBC’s US partner CBS News reports.
However, the FBI says it could not immediately ascertain what type of firearm the gunman used or how many shots were fired.
A Secret Service sniper returned fire and killed the gunman, the agency said.
Footage later shows armed officers approaching a body on the roof of the building.