I am a little muddled and am hoping I can get some clarification about forms in a complex manifold. Since I am only concerned with local issues, consider $M = \mathbb C^n$ as a complex manifold. So I have complex coordinates $z\_1,\ldots,z\_n$ and corresponding real coordinates $x\_1, y\_1,\ldots, x\_n, y\_n$ with $z\_j = x\_j + i y\_j$. Now I have a canonical complex structure $J$ on $M$ given by $J \partial\_{x\_j} = \partial\_{y\_j}, J \partial\_{y\_j} = -\partial\_{x\_j}$. Then I have an isomorphism of the complex bundles between $TM$ and the holomorphic tangent bundle $T\_{(1,0)} M = span\_{\mathbb C} \{\partial\_{z\_j}\}$. This isomorphism is given by
$$
(a+ib) \partial\_{z\_j} \mapsto a\partial\_{x\_j} + b\partial\_{y\_j}.
$$
Now let $\omega = \sum\_i dx\_i \wedge dy\_i$ be the standard symplectic form on $M$. What does this correspond to under the above isomorphism. At first glance it is
$$
\tilde \omega = \frac{i}{2}\sum\_j dz\_j \wedge d\bar z\_j
$$
where $dz\_j = dx\_j + i dy\_j, d\bar{z\_j} = dx\_j - i dy\_j$. At second glance this can't be correct since this is zero on the holomorphic tangent bundle. On third glance, everything seems to work out if I think of
$$
d\bar{z\_j}(c\partial\_{z\_j}) = \bar c.
$$
But this is unsettling to me. Is this the right way to think about it though? Can I get in any trouble by thinking of it this way?

The problem seems to be that the differentials of the real coordinates give real dual vectors, but the covectors $dz\_j$, $d\bar z\_j$ are inherently complex (their real span is not the real dual to $T\_{(1,0)} M$ unless I interpret $d\bar z\_j$ as mentioned above).

I believe what is usually done is $\tilde\omega$ is considered to be a form on $TM \otimes \mathbb C$ and is a real symplectic form on
$$
\mathbb Rspan\{\partial\_{x\_j}, \partial\_{z\_j}\} = \mathbb Rspan\{\partial\_{z\_j} + \partial\_{\bar z\_j}, i(\partial\_{z\_j} - \partial\_{\bar z\_j})\}
$$
but I'd prefer to not complexify if I don't have to (since I am just concerned with the symplectic geometry and thus the real bundle, but I would like the convenience of the complex notation).