linear algebra - How to prove that an invertible matrix is a product of elementary matrices?

Let $A = (a_{ij})$ be an invertible $n\times n$ matrix. I wonder how to prove that $A$ is a product of elementary matrices. I suspect that we need to transform it into the identity matrix by using elementary row operations, but how to do it exactly?

P.S. I've checked questions which could be considered similar and neither of them deals with this exact (general) situation. Please don't mark this question as a duplicate unless you find a precise answer.