There are very few universities that are universally known by an abbreviation. MIT is one of them. Caltech is arguably another, but I find blank faces on the East Coast of the US when I mention the name, so there is an issue of recognition there to begin with.
USC is another that is ambiguous. It's easy to assume if you are from the West Coast that "clearly" this is the University of Southern California, but New Yorkers may or may not recognize this.
New Yorkers tend to think NYU is one of the great institutions in the world, but I find most people on the West Coast aren't familiar with it and default to thinking "NYU" means the state university for New York.
Some are suggesting it's weird to spell things out, but I've seen enough resumes to recognize that people spell out all kinds of things and as long as it's clear what is meant, there is generally more to worry about in a resume than that.
So what about your example, UCLA? I've met very well-educated people from abroad that didn't actually know what it was. It's true the brand is very strong (especially in Asia), but it definitely does not have the recognition of MIT. In this case, you might want to put "University of California, Los Angeles" but you run the risk of some people not recognizing this is the same as UCLA (I've seen this happen with people from Asia). This example shows it's important to recognize your audience.
In summary, whether to use an abbreviation depends heavily on audience. Lean towards not using an abbreviation if you are unsure, and keep in mind very few institutions are known universally to begin with and known by their abbreviation.