Significance of the n-differential forms I know that it is well defined too say that a differential $k$-form on a manifold M with dimension n is somewhat regarded as a map from the tangent space of the manifold $M$ to $\mathbb{R }$ for each $p\in M,$ then we can define $w$ as the $k$-form where:
$$w=\sum_{i_1,...,i_k=1}^n w_{i_1,...,i_k}du^{i_1}\wedge...\wedge du^{i_k}$$
My question is, what is the geometric significance of the $k$-form defined as above when $w(p)\ne 0?$
 A: Differential forms eat vector fields and spit out numbers (or in a more esoteric language: they live in the cotagent bundle).
The geometric significance of a top form $ω$ (i.e. an $n$-form in a $n$-dimensional manifold)is somewhat straight forward: Given $n$ vectors $u_1,...,u_n$ on $T_pM$, 
$w(u_1,...,u_n)=c_p\cdot \det(u_1,...,u_n)$=
$c_p\cdot Vol(\text{the siplex having as edges the vectors $u_1,...,u_n$})$
$c_p$ is just a number  which varies smoothly on the point $p$. 
Now a $k$-form on an $n$ dimensional manifold (not necessarily a top form) does almost the above but before spitting out the volume it projects. So say for example that we have the form $ω(a,b,c)=3(a+b)dx\wedge dy$ on $\mathbb{R^3}$ and we want to compute what it does on $u_1,u_2 \in T_{(1,0,0)}\mathbb{R^3}$. 
$ω$ would take the 2 vectors, project them on the $x,y$ plane and compute the volume of the parallelogram spaned by these 2 projections. Finally it would multiply the volume by some factor, in our case $3(1+0)=3$. 
By analogy, if we had $ω(p)=f(p)dxdy + g(p)dydz+h(p)dzdx$ on $\mathbb{R^3}$ then for any two vectors in the tangent space at the point $p$ it would add the scaled volumes of the parallelograms of the projections at the planes $xy$,$yz$ and $xz$ respectively.
To see why this happens you can check David Bachmann's really good introductory book "A geometric approach to Differential forms" it really is a very nice introduction to the subject with lots of excerises and examples.  
