Elite is the game that started it all, over 20 years ago. I played it on a 48 k Spectrum PC, but it was written on a BBC computer and adapted for virtually all PCs back then, even the Apple IIe. You started with a basic ship, poorly armed and very vulnerable. You had to trade to buy equipment to upgrade your ship, but could also mine, bounty hunt or even be a pirate. There was a whole universe to explore, plus a few missions to do for Her Majesty's Galactic Navy. There were pirates, police who were too easy to get angry at you and an alien enemy to watch out for. There had never been an open ended game like it before, and it inspired many more games -- including eventually Freelancer. There were sequels to Elite, but none was as good.
Then there were the Privateer games, which again had you freelancing and fighting bad guys and aliens. Lots of fun at the time, though the graphics look poor in comparison to Freelancer.
One I liked very much came out about 6 years ago and can be bought for a song on eBay now. It's called HardWar and is sort of like Freelancer on a planet. Again, you have a simple ship to start with and your choice of how to make money. There is also an alien mystery to solve. The idea was to solve the mystery and get off of Titan, one of the moons of Jupiter, where your people had been isolated from Earth for centuries. There were supposed to be sequels, but unfortunately the game didn't sell as well as it should have and there haven't been any.
A game similar to Freelancer is X-Beyond the Frontier and its folow ups. It's a nice enough game, but is very slow and takes forever to make enough money to upgrade your ship. Good graphics though.