As Bruce Westubury noticed, the answer to this question is trivial as it is stated.

Surprisingly enough, however, the situation becomes very interesting when one considers representations of *homogeneous* polynomials as pfaffians of matrices with *linear* entries.

More precisely, let us consider the following version of the question:

**QUESTION.** Let $F \in k[x_0, \ldots, x_n ]$ be a homogeneous polynomial of degree $d$.
Does there exist a symmetric [resp. antisymmetric] matrix $M$, whose entries are linear forms, such that
$$\det(M)=F \quad (\textrm{resp}. \ \textrm{Pf}(M)=F)?$$

This problem is studied in detail in Beauville's paper [Symmetric determinantal hypersurfaces, Michigan Mathematical Journal 48 (2000)], where the existence of a symmetric or pfaffian representation is related to the existence of certain vector bundles on the projective variety $X \subset \mathbb{P}^n_k$ defined by $F=0$.

Among other things, Beauville proves the following results (when $k= \mathbb{C}$):

$\bullet$ A general polynomial of degree $d$ admits a determinantal representation if and only if $$n=2 \quad \textrm{or}$$ $$n=3 \quad \textrm{and} \quad d\leq 3.$$

$\bullet$ A general polynomial of degree $d$ admits a pfaffian representation if and only if $$n=2,$$ $$n=3 \quad \textrm{and} \quad d \leq 15 \quad \textrm{or}$$ $$n=4 \quad \textrm{and} \quad d \leq 5.$$