Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries?

Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries?

The pfaffian can be defined as $\sqrt{{\rm det}(A) } $ when $A$ is skew symmetric, or explicitly $${\rm pf}(A) = \frac{1}{2^n n!}\sum\limits_{\sigma \in S_{2n}}{\rm sgn} (\sigma)\prod\limits_{j=1}^n a_{\sigma (2j -1 ),\sigma(2j)}.$$

Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian?

Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries?