This variety of algebras is well studied. In particular, the description of the underlying vector space of the free algebra in this variety follows immediately from

"A note on the T-ideal generated by $s_3(x_1,x_2,x_3)" by G. D. James, Israel Journal of Mathematics, 29 (1978), 105–112.

The main result of that paper, when translated to free algebras, says that the degree $d$ component of the free algebra generated by $V$ is the following Schur functor of $V$:

$S^{2}(V)\oplus S^{1,1}(V)$ for $d=2$

$S^{4}(V)\oplus 2S^{3,1}(V)\oplus S^{2,2}(V)$ for $d=4$

$S^{d}(V)\oplus 2S^{d-1,1}(V)$ for $d=3$ or $d\ge 5$

This description, in my opinion, is sufficiently pleasant to have good intuition of what these algebras are, present them by generators and relations, write down convenient bases etc.

This variety of algebras is well studied. In particular, the description of the underlying vector space of the free algebra in this variety follows immediately from

"A note on the T-ideal generated by $s_3(x_1,x_2,x_3)$" by G. D. James, Israel Journal of Mathematics, 29 (1978), 105–112.

The main result of that paper, when translated to free algebras, says that the degree $d$ component of the free algebra generated by $V$ is the following Schur functor of $V$:

$S^{2}(V)\oplus S^{1,1}(V)$ for $d=2$

$S^{4}(V)\oplus 2S^{3,1}(V)\oplus S^{2,2}(V)$ for $d=4$

$S^{d}(V)\oplus 2S^{d-1,1}(V)$ for $d=3$ or $d\ge 5$.

This description, in my opinion, is sufficiently pleasant to have good intuition of what these algebras are, present them by generators and relations, write down convenient bases etc.