Let $V$ be a complex finite-dimensional vector space. Then there always exists an isomorphism $V \simeq V^*$, where $V^*$ is the dual space. The isomorphism can be fixed by choosing a non-degenerate bilinear or sequilinear form on $V$: $$\langle \dot \,, \dot \, \rangle : V\times V \rightarrow \mathbb{C}$$ Now, let $G$ be a Lie group acting on $V$ (label this representation $\rho$). Then the dual representation $\rho^*$ to $\rho$ should preserve the bilinear\sequilinear form defined above: $$ \langle{\rho}^*(g)\varphi, \rho(g)v\rangle = \langle\varphi, v\rangle. $$
Taking a sequilinear form seems to be more natural than a bilinear one (since $V$ is complex).
Do the two form types give different dual representations of $G$?
If yes, then which one is usually referred to as the dual representation?