Let $F$ be a field and $A$ an $F$-algebra with an antisymmetric product $\cdot$. In other words, for all $v,w\in A$ we have $$v\cdot w=-w\cdot v.$$ Examples of such algebraic object include all Lie algebras.
Question: Is there an example that is not a Lie algebra?
The only thing this algebra could miss is the Jacobi identity: $$u\cdot(v\cdot w)+v\cdot(w\cdot u)+w\cdot(u\cdot v)\neq 0,$$ for some $u,v,w\in A$.
For an explicit example, take the $3$-dimensional Heisenberg Lie algebra with basis $(e_1,e_2,e_3)$ and products $e_1\cdot e_2=-e_2\cdot e_1=e_3$ and all other products zero. Now introduce in addition the products $e_1\cdot e_3=-e_3\cdot e_1=e_1$. Then the algebra product is still skew-symmetric, but does no longer satisfy the Jacobi identity: $$ (e_1\cdot e_2)\cdot e_3 + (e_2\cdot e_3)\cdot e_1 + (e_3\cdot e_1)\cdot e_2 =-e_3. $$