"a lot of space" is correct and usable in written English. You can use it when referring to physical space, literal or figurative space, or the concept of having plenty of time or opportunity to do something. For example, "We have a lot of space in our new office, allowing us to spread out and work comfortably.".
Elephants need a lot of space".
She gives this a lot of space.
I don't have a lot of space.
"There's a lot of space for everyone".
We've got a lot of space here".
"There's a lot of space around here.
You have a lot of space.
I love the desktop app, it’s always running on my Mac. Ludwig is the best English buddy, it answers my 100 queries per day and stays cool.
Cristina Valenza
Retail Lead Linguist @ Apple Inc.