As a thought experiment, suppose we live in an alternative world where the progress in mathematics is identical to ours, except that so far nobody has come up with the determinant $\det: GL_n(\mathbb{R}) \to \mathbb{R}^{\times}$ given by the Leibniz formula
$A \mapsto \sum_{\sigma \in S_n} \left( \text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma_i}\right)$, for $A \in GL_n(\mathbb{R})$.
Suppose we've nevertheless already figured out that $GL_n(\mathbb{R})$ has exactly two connected components, by noticing that $GL_n(\mathbb{R})$ deformation retracts to $O_n(\mathbb{R})$, and that $\pi_{0}(O(n)) \cong \mathbb{Z}_2$.
Let us be interested in classifying $T \in GL_n(\mathbb{R})$ by finding an explicit map $\phi: GL_n(\mathbb{R}) \to \mathbb{Z}_2$ sending $T$ to $\bar{0}$ if $T$ lies in the identity component, and to $\bar{1}$ otherwise.
In this hypothetical setting, by only studying the properties of $\phi$, would we be able to discover the Leibniz formula?
Will every such $\phi$ necessarily turn out to factor through $\det$ as $\phi = \alpha \circ \det$ for some $\alpha: \mathbb{R}^{\times} \to \mathbb{Z}_2$ ? If so, what will the collection of all possible $\alpha$ look like? Otherwise, what would be an example of an alternative for $\phi$?