Let $\alpha,\beta$ be two $k$-forms on a manifold $M$. If there exists some chart $(U,x^1,\dots,x^n)$ on which $\alpha=\beta$ does it follow that $\alpha$ and $\beta$ are the same forms? In different charts $\alpha$ and $\beta$ look different; say that on $U$ $$\alpha=\alpha_{i_1,\dots,i_k}dx^{i_1}\wedge\dots\wedge dx^{i_k}$$ and $$\beta=\beta_{i_1,\dots,i_k}dx^{i_1}\wedge\dots\wedge dx^{i_k}.$$ If these two local expressions are equal, why do their representations in all other local coordinates agree?
My question arises as I can show two things are equal in normal coordinates, and I want to conclude that these two things are thus equal in a coordinate free sense.
Not if this is true on only one chart. For example, consider the 1-form metrically dual to the north-south vector field on the round sphere. Now multiply it by a bump function supported away from a small neighborhood of the north pole to produce a second 1-form. The two 1-forms agree away from that neighborhood. In particular, they agree on a chart covering the northern hemisphere. However, they are not globally the same.
Now, if you could show that they agree on a collection of charts that cover $M$, then you can conclude that they are equal.
(This is because one-forms are smooth, not analytic. Loosely speaking, for analytic objects, agreement on an open set implies global agreement. For smooth objects, agreement on an open set implies nothing more.)