How does one express $\theta_{i_1\dots i_k}$ as a linear sum of $\theta_{i_{1'}\dots i_{k'}}$?

Question

Let $\{dx^{i_1}\wedge\cdots\wedge dx^{i_k}\mid 1\leq i_1<\cdots<i_k\leq n\}$ and $\{dx^{i_1'}\wedge\cdots\wedge dx^{i_k'}\mid 1\leq i_{1'k}<\cdots<i_{k'}\leq n\}$ be two basis for the collection of anti-symmetric $\begin{pmatrix}0\\k&\end{pmatrix}$ tensors. Let $\theta$ be one of its element. Expand it w.r.t. the first basis shown above, and we get the components , where $1\leq i_1\dots i_k\leq n$. Similarly, we get $\theta_{i_{1'}\dots i_{k'}}$. What is the relationship between $\theta_{i_1\dots i_k}$ and $\theta_{i_{1'}\dots i_{k'}}$? In other words, how does one express $\theta_{i_1\dots i_k}$ as a linear sum of $\theta_{i_{1'}\dots i_{k'}}$?