I am sorry for this type of question, but I currently have to deal with differential forms although I have not heard so far what they actually are, so I have just a few very particular questions about them: So let $M$ be a manifold.
1.) What is a 2-form?
Wikipedia tells me that a 1-Form is a map: $\omega : TM \rightarrow \mathbb{R}$ such that $\omega(p,.) \in T_p^*M.$
This is pretty understandable.
Now, I assume that a two-form is a map $\omega: TM \times TM \rightarrow \mathbb{R}$ such that $T(q,.,x,y) \in T_q^*M$ and $T(r,s,a,.) \in T_a^*M$ for all $a,x,y,r,s,q$, is this correct? If not, could you give a definition with most simple notation?
2.) When do we call a 2-form non-degenerated?
All I can offer here is this wikipedia article Degenerate form, but it does not talk much about 2-forms, I think.
3.) What are these $dx_i$ guys?
What I read is the following: If you have a coordinate system (collection smooth maps $x_i : M \rightarrow \mathbb{R}$) on your manifold, then the total differentials $dx_i(p)$ form a basis of $T_p^*M,$ since we wanted for a 1-form that $\omega(p,.) \in T_p^*M,$ there is a representation of this linearform with respect to the $(dx_i(p))_i.$
Despite, I have some issues with this(in case that this is true): What is the precise definition of a coordinate system?
When do we call a map from a manifold smooth?
A 2-form is a smooth choice of a skew-symmetric bilinear form on each of the tangent spaces of your manifold. A 2-form is non-degenerate if it is non-degenerate (as a bilinear form) when restricted to each tangent space.
For a general introduction to differential forms and smooth manifolds see Lee's Introduction to Smooth Manifolds. For a general introduction to the linear algebra of skew-symmetric non-degenerate bilinear forms see chapter 2 of McDuff-Salamon's Introduction to Symplectic Topology.