The Jacobi identity in a Lie algebra $\mathfrak{g}$ looks like this (for $x,y,z\in \mathfrak{g}$): $$[[x,y],z]-[x,[y,z]]+[y,[x,z]]=0.$$ This just says that the map $y\mapsto [x,y]$ is a derivation with respect to the Lie bracket $[-,-]$.
It also looks like some sort of cocycle condition. An alternating sum equal to zero. Can this be made precise? Is the Jacobi identity the kernel of some differential?
For the adjoint $\mathfrak{g}$-module $\mathfrak{g}$ we have $$ Z^1(\mathfrak{g},\mathfrak{g})={\rm Der}(\mathfrak{g}),\quad B^1(\mathfrak{g},\mathfrak{g})={\rm ad}(\mathfrak{g}) $$ for the Lie algebra cohomology. So the map ${\rm ad}(x)$, defined by $y\mapsto [x,y]$ is a $1$-cocycle for the Lie algebra cohomology with coefficients in the adjoint module. In this sense, the Jacobi identity is a derivation condition, which is a $1$-cocycle condition.
In general, the space of Lie algebra $1$-cocycles and $1$-coboundaries with coefficients in a $\mathfrak{g} $-module $M$ is given by \begin{align*} Z^1(\mathfrak{g},M) & = \left \{ \omega \in Hom (\mathfrak{g},M) \mid \omega ([x,y])= x\cdot \omega (y)-y\cdot \omega(x) \right \}\\ B^1(\mathfrak{g},M) & = \{ \omega \in Hom (\mathfrak{g},M) \mid \omega (x)=x\cdot m \mbox{ for some } m\in M \} \end{align*}