The best indie games like Skate 3 are Session: Skate Sim, Skater XL and OlliOlli World. Below are 6, each with the reason it is here — and, at the end, what filtering to indie costs you.
Filtering to indie means you want the answer a small team would give. It is usually the most interesting filter on this site, because small studios take the risks large ones cannot.
What you get: One strong idea, explored completely, at a price that makes trying it easy. Indies have no production to hide behind, so the design is exposed.
What made Skate 3 worth playing in the first place is covered on the full list. This page is the indie part of that answer, and only the indie part.
The most committed successor — a stick per foot, and no score at all.
Flick-it taken further, with an enormous mod scene.
Skating as a 2D precision platformer, and beautifully made.
Skating and shooting, with a stunning stylised look.
Downhill on a bike, procedural, and all about flow.
Grinding a city with the style turned all the way up.
Polish and scale, sometimes. A few of these are rough at the edges, and a big-budget pick will always look better doing the same thing.
Specifically, these are strong answers for Skate 3 that are not indie games, so they are not on this page:
The full list is on games like Skate 3 — 9 picks, no genre filter.
Session: Skate Sim. The most committed successor — a stick per foot, and no score at all.
No. Skate 3 released on the PlayStation 3, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Self-published, or published without a large publisher's money. It says nothing about the size of the game, which is why a forty-hour metroidvania and a two-hour art piece sit under the same label.
Constraint. Without a budget to hide behind, one idea has to carry the whole thing — and a small team can commit to something risky that a large one would sand down.
By hand. Each pick was written for the full Skate 3 list first, on what the game actually does, and this page is the part of that list that is indie — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the indie guide for where to start in the genre generally.