Cross product is not associative, but is there any non-trivial associative algebra that has all the algebraic properties of cross product except the condition that would violate associativity and associativity? It would be preferrable if numbers in the algebra are vectors in $\mathbb{R}^3$.
By algebraic property of cross product, I mean things like anti-commutativity, $\mathbf{a} \times \mathbf{a} = 0$, distributivity over addition.
A lie algebra $\frak g$ is a vector space equipped a bilinear operation $[-,-]$ satisfying $[x,x]=0$ for all $x\in\frak g$ and the Jacobi identity $[x,[y,z]]+[z,[x,y]]+[y,[z,x]]$. The space $(\Bbb R^3,\times)$ is a lie algebra.
One should look up the classification of three-dimensional real lie algebras (which is fairly easy and textbook, but I don't have it memorized or anything) to see if there are any associative ones, besides the trivial abelian lie algebra of course (where the operation is $[v,w]=0$ for all $v,w\in\Bbb R^3$).
What you seem to want is a lie algebra whose bracket operation is associative. Usually it is not associative. In fact, $\frak g$ is associative iff its derived subalgebra is contained in its center.
Here's an example of such an algebra: quotient the full exterior algebra on a vector space by the relations encoding the Jacobi identity (we can restrict ourselves to only a basis of the space). This should be a "universal" construction in a sense. Indeed, if $\frak g$ is an associative Lie algebra of dimension $n$, then it should a homomorphic image of the lie algebra
$$\frac{\Bbb C\langle x_1,\cdots,x_n\rangle}{(x_ix_j+x_jx_i,\, x_ix_jx_k+x_kx_ix_j+x_jx_kx_i)_{1\le i,j,k\le n}},$$
where $R\langle X\rangle$ stands for the nonabelian polynomial ring over $R$ generated by a set $X$, and the bracket operation is just multiplication.