Someone comments "this is exactly what we need" under your LinkedIn post. The tempting move is to send a proposal that night. It's usually too early. You don't know the budget, the deadline, or who signs off.
Ask two or three questions in a DM first. What is the problem costing them right now? When does it need to be done? Who else has to say yes?
If they answer, write a one-page brief back in their words and ask them to correct it. The proposal comes after they agree the brief is right.
This guide on freelancer social selling on LinkedIn walks through it, from the proof posts to the written brief.