Is there any analogy of symmetric matrix in higher-degree homogeneous polynomials?

Question

In quadric forms, we know that we can use symmetric matrix to describe it, and by diagonalizing it we can determine the isomorphic classes of them as projective varieties (use its rank and dimension, see Wedhorn's AG, the last part of chapter 1). But I want to ask is there any analogy of symmetric matrix for curves of higher degrees in $\mathbb P^n$? If it can be realized, can we reduce the research of curves in $\mathbb P^n$ to "that" object? Thank you.