I want to determine whether a certain n-dimensional, simply-connected vector field is conservative. I can very easily calculate its Jacobian in numerical form.
In 3 dimensions, the problem is easy: the Jacobian is a 3x3 matrix, and the curl can be obtained from its elements:
[jacobian[2,1] - jacobian[1,2], jacobian[0,2] - jacobian[2,0], jacobian[1,0] - jacobian[0,1]]
A curl of zero would mean that the vector field is conservative.
But what if my Jacobian has 100 elements? The curl is only defined for 3 dimensions, but is there a formula for some high-dimensional generalization?
On a simply connected region of $\mathbb{R}^n$ the condition is that, in your notation, jacobian[i,j]=jacobian[j,i] for any i,j. This generalizes your curl condition.
Explanation: your vector field corresponds to a $1$-form $\omega$, and the vector field is conservative if and only if $\omega=d\phi$ for some $0$-form $\phi$. If the base is simply connected, then this is equivalent to $d\omega=0$. Now $d\omega$ is a $2$-form, and the local coordinates with respect to the standard basis are the differences jacobian[i,j]-jacobian[j,i] for distinct i,j.