Fact one: If $V$ is a three-dimensional vector space, consider the matrix representation $A$ of a linear transformation from $V$ to $V$. Then the characteristic polynomial of the $3 \times 3$ matrix $A$ is given by $$ p(x) = x^3 - (\operatorname{tr} A)x^2 + \frac{1}{2}(\operatorname{tr}(A)^2 - \operatorname{tr}(A^2))x - \det A \,. $$
Fact two: If $V$ is a finite-dimensional vector space and $G$ is a finite group, we can take a representation $\rho \colon G \to \operatorname{GL}(V)$. This representation has a character; call it $\chi_V$. Additionally, we have the induced 2-tensor representation, which has character $\chi_{V \otimes V}(g) = \chi(g)^2$. One can show that the subrepresentation $(G, \Lambda^2 V)$ of $(G, V \otimes V)$ has character $\chi_{\Lambda^2 V}(g) = \frac{1}{2}(\chi(g)^2 - \chi(g^2))$.
My question: What is the explicit connection between the linear term of the characteristic polynomial above and the character formula for $\chi_{\Lambda^2 V}(g)$? Obviously characters are traces in this finite-dimensional setting, but I am for some reason having a hard time seeing what exactly the connection is. Thanks!