In dictionaries we use, There is formal and informal written at the side of some words. Most of the words have no remark like that.
Can we use those words in our formal writings?
1) Ask for some thing. - Jodie asked for a guitar for her birthday.
2) Grapple with something. - This is an issue that is being grappled with by the council.
3) Nail somebody down - He says he will come, but I can't nail him down to a date.
These are the examples from Oxford Dictionary, Indian version.
You should be safe to assume that unlabelled words and phrases could be used both formally and informally.
Having said that, #3 above is definitely informal and #2 is borderline in my opinion.