I just read section 4 of the manifold book by Loring Tu.
I can understand the $1$-form without questions as covector fields. A $k$-form on $U$ is a function that assigns every $p\in U$ an alternating $k$-linear function on $T_p(\mathbb{R}^n)$. I understand that we need to assign a $k$-linear function for integration to make sense. However, why do we need the alternating instead of just $k$-linear function?
Suppose $O, U \subset \mathbb{R}^n$ and $F : O \to U$ is a diffeomorphism, to be interpreted as a change of coordinates from $x \in U$ to $y \in O$. We integrate alternating forms because the transformation law for coordinate transformation $x = F(y)$ of an $n$-form $\alpha = a(x)dx_1 \wedge \dots \wedge dx_n$ on an open set $U \subset \mathbb{R}^n$ reads $$F^*(a(x)dx_1 \wedge \dots \wedge dx_n) = a(F(y))\det DF(y) dy_1\wedge \dots \wedge dy_n,$$ while the change of variables formula for integration reads $a(x)dx = a(F(y)) |\det DF(y)|\,dy$. These formulas say that $\int_{U}\alpha := \int_{U}a(x)\,dx$ is independent of the coordinate system, provided that $\det DF > 0$, which is to say that $F$ preserves orientation.
This construction wouldn't work for a general $n$-tensor $A = a(x)dx_1 \otimes \dots \otimes dx_n$ that is not alternating because the transformation formula no longer matches the change of variables theorem.