I have a $3 \times 3$ matrix, say:
$$ A = \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix}. $$
I was wondering if there is a name for the quantity: $$ (ei-hf)+(ai-gc)+(ae-db), $$ which is the sum of the three $2\times 2$ determinants calculated with respect to $a, e, i$, respectively.
Expanding the determinant of the $3 \times 3$ matrix $t I - A$ gives $$\det(t I - A) = t^3 - (\underbrace{a + e + i}_{\operatorname{tr} A}) t^2 + (\underbrace{ae + ai - bd - cg + ei - fh}_{\sigma_2(A)}) t + (\underbrace{aei - \cdots}_{\det A}) .$$ We can see immediately that the quantity you're interested in is the linear coefficient, $$\sigma_2(A) = ae + ai - bd - cg + ei - fh,$$ and so it's a special case of a natural generalization of the trace and determinant.
We can put this another way: Diagonalizing gives that the determinant of $t I - A$ can also be expressed in the eigenvalues $\lambda_a$ of $A$ as $$\det(t I - A) = \prod (t - \lambda_a) = t^3 - (\underbrace{\lambda_1 + \lambda_2 + \lambda_3}_{\operatorname{tr} A}) t^2 + (\underbrace{\lambda_2 \lambda_3 + \lambda_3 \lambda_1 + \lambda_1 \lambda_2}_{\sigma_2(A)}) - \underbrace{\lambda_1 \lambda_2 \lambda_3}_{\det A} :$$ $\sigma_2(A)$ is the second elementary symmetric polynomial in the eigenvalues of $A$, just as (up to sign, and for $3 \times 3$ matrices) the trace and determinant are respectively the first and third elementary symmetric polynomials in the eigenvalues.
With this viewpoint in hand, we can also use Newton's Identities and the fact that the eigenvalues of $A^2$ are $\lambda_a^2$ to write this quantity as $$\sigma_2(A) = \tfrac{1}{2}[(\operatorname {tr} A)^2 - \operatorname{tr} (A^2)] .$$