I'm reading "Differential forms in algebraic topology" by Bott. In chapter 1 they introduce differential forms as elements of $\Omega^*(\mathbb{R})=\{C^{\infty}$ functions on $\mathbb{R^n}\}\otimes\Omega^*$
$\Omega^*$ is the algebra over $\mathbb{R}$ generated by $dx_1,...,dx_n$ with the relations $(dx_i)^2=0$ and $dx_idx_j=-dx_jdx_i$. $\Omega^*$ has basis $1,dx_i, dx_idx_j,dx_idx_jdx_k,...,dx_1...dx_n$ (note $i<j<k$).
Therefore $\omega=\sum f_{i_1...i_k}dx_i...dx_k$ with $f_{i_1...i_k}$ are $C^{\infty}$ functions and also write $\omega=\sum f_I dx_I$ ($I$ increasing index sequence).
In an example they say that $\omega\in\Omega^1(\mathbb{R^3})$ has the form $f_1dx_1+f_2dx_2+f_3dx_3$ and $\in\Omega^2(\mathbb{R^3})$ has the form $f_1dx_2dx_3-f_2dx_1dx_3+f_3dx_1dx_2$.
How to I get this shape of $\omega$ from the above statements? For $\omega\in\Omega^2(\mathbb{R^3})$ why do the indices of $f_i$ differ to the indices of the $dx_jdx_k$ and why do we get a minus if we have an odd permutation of the indices. Thanks for your help.