Complex differential forms and the determinant formula I couldn't find this on the literature, but in some simple cases it is possible to compute differential complex forms using the determinant formula, just as the real case. One simple example is given by $$dz_k \wedge d\overline{z_l} \left(\frac{\partial}{\partial z_k}, \frac{\partial}{\partial \overline{z_l}}\right) = \det \left[
\begin{array}{cc}
   dz_k\left( \frac{\partial}{\partial z_k} \right) & d\overline{z_l}\left( \frac{\partial}{\partial z_k} \right)\\
dz_k\left( \frac{\partial}{\partial \overline{z_l}} \right) & d\overline{z_l}\left( \frac{\partial}{\partial \overline{z_l}} \right) \end{array} \right] = 1.$$
I wonder if this is valid in general. More precisely, is valid the (more general) formula below?
$$dz_{i_1} \wedge \ldots \wedge dz_{i_p} \wedge d\overline{z_{j_1}} \wedge \ldots \wedge d\overline{z_{j_q}}(v_1, \ldots, v_{p+q}) = $$
$$ = \det \left[
\begin{array}{cccccc}
   dz_{i_1}(v_1) & \ldots & dz_{i_p}(v_1) & d\overline{z_{j_1}}(v_1) & \ldots & d\overline{z_{j_q}}(v_1)\\
\vdots & & \vdots & \vdots & & \vdots\\
dz_{i_1}(v_{p+q}) & \ldots & dz_{i_p}(v_{p+q}) & d\overline{z_{j_1}}(v_{p+q}) & \ldots & d\overline{z_{j_q}}(v_{p+q})\\ \end{array} \right]$$
Thank you for you help.
 A: I believe this determinant formula is quite often taken as the definition of the wedge product of differential forms. It doesn't matter whether they are complex or real here. However one should be careful, because sometimes wedge product with different normalization is used, and your formula should then be modified by some combinatorial factor.
EDIT
Let $\omega = \omega^1 \wedge ... \wedge \omega^n$ be an $n-$form constructed as wedge product of $1-$forms $\omega^i$. Then by definition of the wedge product we have
$$ \omega(X_1...X_n) = \mathrm{det} \left( \omega^i ( X_j )  \right)_{i,j=1...n}. $$
Note that I didn't use any coordinates or particular basis to write this formula down. Every form can be written down as a sum of product forms as above, so action of every $n$-form can be evaluated. Sometimes it is introduce to (locally) introduce a basis on the tangent and cotangent planes. We can pick arbitrary vector fields $e_i$ and $1-$ forms  $f^j$ such that $$ f^j (e_i) = \delta^j_i .$$
Often one uses some specific coordinates $x^i$ and picks $e_i = \frac{\partial}{\partial x^i}$, $f^j = d x^j$. Note however, that there is nothing special about this choice - any other will do. For example on a complex manifold we use $dz^i$ and $d \bar z^i$, even though they surely can't be regarded as independent real coordinates.
