The answer is no.

In fact, let $\Gamma \subset \mathbb{P}^3$ be the set of $10$ nodes of a general quartic symmetroid. Then the Gale transform $\Gamma' \subset \mathbb{P}^5$ of $\Gamma$ must be contained in a Veronese surface.

Another characterization is the following: the set $\Gamma$ arises as a linear section of the secant variety $\textrm{Sec}(C_6)$, where $C_6 \subset \mathbb{P}^6$ is a rational normal curve. Note that $\textrm{Sec}(C_6)$ has precisely degree $10$, since $10$ is the number of nodes of a general projection of $C_6$ to $\mathbb{P}^2$.

For (many) more details, see

D. Eisenbud, S. Popescu: The projective geometry of the Gale transform, Journal of Algebra 230 (2000), 127-173,

in particular Example 6.3 p. 153.

The answer is no.

In fact, let $\Gamma \subset \mathbb{P}^3$ be the set of $10$ nodes of a general quartic symmetroid. Then the Gale transform $\Gamma' \subset \mathbb{P}^5$ of $\Gamma$ must be a zero dimensional scheme of length $10$ given by the simple intersection points of two Veronese surfaces, and this condition is not satisfied by a collection of $10$ general points.

Another characterization is the following: the set $\Gamma$ arises as a linear section of the secant variety $\textrm{Sec}(C_6)$, where $C_6 \subset \mathbb{P}^6$ is a rational normal curve. Note that $\textrm{Sec}(C_6)$ has precisely degree $10$, since $10$ is the number of nodes of a general projection of $C_6$ to $\mathbb{P}^2$.

For (many) more details, see

(1) D. Eisenbud, S. Popescu: The projective geometry of the Gale transform, Journal of Algebra 230 (2000), 127-173,

in particular Example 6.3 p. 153.

(2) D. Eisenbud, K. Hulek and S. Popescu: A note on the intersection of Veronese surfaces, Commutative Algebra, Singularities and Computer Algebra, Volume 115 of the series NATO Science Series (2003), pp 127-139,

in particular Section 3.