Suppose:
$$ \begin{align} \mathbf{e}_1&:=a\gamma_1+b\gamma_2\\ \mathbf{e}_2&:=c\gamma_1+d\gamma_2 \end{align} $$
It is known that the wedge product $\mathbf{e}_1\wedge\mathbf{e}_2=\sqrt{|g|}\gamma_1 \wedge \gamma_2$ has a 'nice' expression referencing the metric tensor and the determinant.
I suspect that we can write any basis element in any combinations of wedge products in terms of the metric tensor.
Specifically, can we write $\mathbf{e}_1$ in terms of $\gamma_1$ and $\gamma_2$ and the metric tensor $g$?
For instance, can we write $\mathbf{e}_1$ as an expression along these lines
$$ \mathbf{e}_1\approx g^{\mu\nu} \gamma_\nu $$
(I know this equation is not the correct one, but I can't figure it out - hence this request for help).
Edit:
What about a linear combination of $\mathbf{e}_1$ and $\mathbf{e_2}$ --- does it have an expression in terms of the metric tensor? For example:
$$ F\mathbf{e}_1+G\mathbf{e}_2=H(g,\gamma_1,\gamma_2) $$
Edit 2 / attempt:
$$ \begin{align} \mathbf{e}_1+\mathbf{e}_2&= a\gamma_1+b\gamma_2 + c\gamma_1+d\gamma_2\\ &=(a+c)\gamma_1+(b+d)\gamma_2 \end{align} $$
The metric tensor is :
$$ g=\pmatrix{a^2+b^2 & ac+db\\ ac+bd & c^2+d^2} $$