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 href="https://link.springer.com/article/10.1007/BF02760404">"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</a>.

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.