Is there any basic example about how to understand the area and volume form? From where do we have this volume form?
Is there any way to relate this volume form with a very basic example? Is it possible to relate this volume form to the parallelepiped's volume?
On
In the same vein, a basic $3$-form is a map which is defined by $$ dx^i\wedge dx^j\wedge dx^k(u,v,w) = \det\left(\begin{array}{ccc} d x^i(u) & d x^i(v) & d^i(w) \\ \\ d x^j(u) & d x^j(v) & d^j(w) \\ \\ d x^k(u) & d x^k(v) & d^k(w) \end{array}\right).$$ This represents the volume of the projections $i,j,k$ of the components of vectors $u,v,w$ in $\mathbb R^n$
For three vector $u,v,w$ in $\mathbb R^3$ this is the triple product $u\times v\bullet w$ classically known to give the volume span by those vectors, as Ernie060 is mentioning.
The most basic and simple idea behind differential forms (at least to me) is that they provide a coordinate free approach to analysis and geometry. As a simple example, forms can be used to define integrals over curves, surfaces, and higher-dimensional manifolds in a uniform way. The wikipedia page can give you some flavour of the usefulness of forms.
Example. Consider the 2-form $\omega = dx \wedge dy$ in $\mathbb{R}^2$. If $v, w \in \mathbb{R}^2$ are two vectors, then $$ \begin{align*} \omega(v,w) = (dx \wedge dy) (v,w) &= dx(v) dy(w) - dx(w)dy(v) \\ &= v_1 w_2 -v_2 w_1 \\ &= \begin{vmatrix} v_1 & w_1 \\ v_2 & w_2 \end{vmatrix}. \end{align*} $$ So $\omega(v,w)$ indeed gives the oriented 2-volume (the area) of the parallelepiped's volume in $\mathbb{R}^2$. A similar calculation can be used to show that $dx \wedge dy \wedge dz(u,v,w)$ is the oriented 3-volume (or just volume) of the parallelepiped in $\mathbb{R}^3$ spanned by $u$, $v$ and $w$.