The short answer, is no.
The more full answer is ‘not really’, but you mostly have to represent a body of some sort whether that’s a country, a territory with its own NOC or another body created by the IOC (e.g. the Unified Team or the Refugee Olympic team).
There are ‘independent athletes’ that have competed when political situations beyond the control of the IOC have intervened (the break-up of Yugoslavia and the dissolution of the Netherland Antilles for example) but this is very rare and under the control of the IOC. It’s not a choice an athlete can simply make.
The long answer, with all the gory details, is below…
The ‘Let’s Look At It From A Whole Load of Angles’ Answer
Not only do you not need to represent a country when you participate at the Olympics Games, on some at least, you never do.
When you compete for glory at the Olympics, you are doing so under the banner of your National Olympic Committee. This NOC, of course, represents...