Why is the space of differential forms $\bigoplus_{p=0}^n \Lambda_x^p$? In Wald's book "General Relativity", the space $\Lambda_x$ of differential forms at a point $x$ is worked out in the following manner: Let $M$ be an $n$-dimensional manifold. The vector space of all $p$-forms at a point $x \in M$ is given by $\Lambda_x^p$. The vector space $\Lambda_x$ of all differential forms (not limited to a specified degree $p$) is given by the direct sum:
$$
\Lambda_x =\bigoplus_{p=0}^n \Lambda_x^p
$$
However, from my understanding, the direct sum $A \oplus B$ of two spaces $A$ and $B$ consists of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$, with the additional structure:
$$(a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2)$$
Wouldn't this mean that an element from $\omega \in \Lambda_x$ would take the form:
$$
\omega = (\omega_1, \omega_2, \dots , \omega_n)
$$
where $\omega_1$ is a 1-form, $\omega_2$ a 2-form, and so on? This to me doesn't seem to be the space of all differential forms at $x$, unless all $\omega_i$ are 0 except one for each $\omega$. In fact, it seems to be a much larger space, since it can contain multiple differential forms of different degrees. My first guess was that a $p$-form and a $q$-form would be combined via the map $ \bigwedge: \Lambda^p_x \times \Lambda^q_x \mapsto \Lambda^{p+q}_x $, but then we are no longer limited to the degree $n$. Why is this written as a direct sum rather than, for example, a union?
$$
\Lambda_x = \bigcup_{p=0}^n \Lambda_x^p
$$

I would consider it as the space of formal sums of differential forms. So an element in the space might be $3dx_1 \wedge dx_2 \wedge dx_3 - 19dx_1 \wedge dx_2 + 4$. The operations like $\wedge$ and addition still make sense if you extend them by linearity.
