$\hat{e}^{2}(\vec{v}) = \hat{e}^{2}(2hat{e}_1 + 3\hat{e}_2) = 2 \hat{e}^{2}(\hat{e}_1) + 3\hat{e}^{2}(\hat{e}_2) = 3$
If your space has a metric/inner-product we could *also* have gotten $3$ by noticing $<\hat{e}^2,\vec{v}> = <\hat{e}^2,2\hat{e}_1 + 3\hat{e}_2> = 3$
So when you have a metric you can exploit the
$<\hat{e}_1,\hat{e}_1> = <\hat{e}_1,\hat{e}^1> = <\hat{e}^1,\hat{e}_1> = <\hat{e}^1,\hat{e}^1>$
symmetry to go between vectors and coveectors as you please, messing around with defining things like $<\hat{e}^1,\hat{e}_2> = \hat{e}^1(\hat{e}_2} = \delta^1_2$