Example 2: Here is an example of an $8$-dimensional subspace $N\subset\Lambda^2(\mathbb{R}^8)$ that has no corresponding $a\in\Lambda^4(\mathbb{R}^8)$.

Obviously, this same technique is going to work for any compact simple Lie group of dimension $8$ or more, so there are lots of examples. There is probably a much simpler example that one can write out more explicitly, but I don't see one off the top of my head.

Added remark: This example generalizes to all of the irreducible representations $H_k$ of $\mathrm{SO(3)}$, where the dimension of $H_k$ is $2k{+}1$.

Example 2: Here is an example of an $8$-dimensional subspace $N\subset\Lambda^2(\mathbb{R}^8)$ that has no corresponding $a\in\Lambda^4(\mathbb{R}^8)$.

Obviously, this same technique is going to work for any compact semisimple Lie group of dimension $8$ or more, so there are lots of examples.

No. I don't know the lowest dimension in which this can happen, but there is an example of an $8$-dimensional subspace $N\subset\Lambda^2(\mathbb{R}^8)$ that has no corresponding $a\in\Lambda^4(\mathbb{R}^8)$.

Here is a construction: Recall that the compact Lie group $\mathrm{SU}(3)$ has dimension $8$ and its de Rham cohomology is nontrivial only in degrees $0$, $3$, $5$, and $8$. Identify $\mathbb{R}^8$ with ${\frak{g}} = {\frak{su}}(3)$, the tangent space of $\mathrm{SU}(3)$ at the identity matrix, and note that, if $\phi\in\Omega^3\bigl(\mathrm{SU}(3)\bigr)$ is a bi-invariant $3$-form representing a nonzero element of $H^3_{dR}\bigl(\mathrm{SU}(3)\bigr)$, then the value of $\phi$ at the identity is the Cartan $3$-form $$ \kappa(x,y,z) = -\beta\bigl(x,[y,z]\bigr), $$ where $\beta:{\frak{g}}\times{\frak{g}}\to\mathbb{R}$ is the Killing form (which is nondegenerate and allows us to identify $\frak{g}$ with $\frak{g}^\ast$ in a natural way). Now set $$ N = \{i_x\kappa\ |\ x\in \frak{g} \}\subset \Lambda^2(\frak{g}^\ast)\simeq\Lambda^2(\mathbb{R}^8), $$ where $i_x$ denotes interior product with $x\in\frak{g}$. Then $N$ has dimension $8$, and it is easy to see, from the multiplication properties of the Lie algebra ${\frak{g}} = {\frak{su}}(3)$, that $\omega^2\not=0$ for any nonzero $\omega\in N$.

No, such an $a$ does not always exist. The lowest dimension in which an example without an $a$ could possibly exist is dimension $5$, and, low and behold, there is such an example. I will give one now, but I will also leave the $8$-dimensional example as part of my answer, since I think it is nice and since it illustrates a general principle.

Example 1: For the $5$-dimensional example, let's identify $\mathbb{R}^5$ with the space $\mathsf{S}$ of traceless, symmetric, $3$-by-$3$ matrices with real entries. For each skew-symmetric $3$-by-$3$ matrix $x$, define a $2$-form $\omega_x$ on $\mathsf{S}$ by the rule $$ \omega_x(s_1,s_2) = \mathrm{tr}\bigl(\ x\ [s_1,s_2]\ \bigr) = \mathrm{tr}\bigl(\ x\ (s_1s_2-s_2s_1)\ \bigr) $$ for all $s_1,s_2\in\mathsf{S}$. The space of such $2$-forms is a $3$-dimensional vector space $N\subset\Lambda^2(\mathsf{S}^\ast)$. It is easy to check that ${\omega_x}^2\not=0$ for all nonzero $x$. However, the cone in $\Lambda^4(\mathsf{S}^\ast)$ consisting of the squares of elements of $N$ cuts every hyperplane in $\Lambda^4(\mathsf{S}^\ast)$, so it does not lie properly on one side of any hyperplane. Thus, there does not exist an $a\in\Lambda^4(\mathsf{S}^\ast)$ such that $\langle a; {\omega_x}^2\rangle>0$ for all nonzero $x$.

A word about the proof: This can be done by hand, just writing everything out in a basis, but there is an easier way: Note that the entire construction is invariant under action by $\mathrm{O}(3)$, i.e., if $A$ is an orthogonal $3$-by-$3$ matrix, then, letting $A$ act on $\mathsf{S}$ by $A\cdot s = A s A^{-1}= AsA^{\mathsf{T}}$ and letting $A$ act on $N$ by $A\cdot \omega_x = \omega_y$ where $y = AxA^{\mathsf{T}}$, one sees that the action of $A$ on $S$ preserves the subspace $N$, acting transitively on the unit sphere in $N$, so either ${\omega_x}^2$ vanishes identically or it is never zero for nonzero $x$. Computing one example shows that the latter holds. Next, the cone of squares of elements of $N$ in $\Lambda^4(\mathsf{S}^\ast)$ must also be invariant under the action of $\mathrm{O}(3)$, and, if it there were an $a$ in this space whose inner product with all of the elements of the cone were nonnegative, then averaging $a$ over the action of $\mathrm{O}(3)$ would yield a nonzero vector $\bar a$ in $\Lambda^4(\mathsf{S}^\ast)$ that was fixed under $\mathrm{O}(3)$. However, $\Lambda^4(\mathsf{S}^\ast)$ is isomorphic to $\mathsf{S}$ as an $\mathrm{O}(3)$-module and hence is irreducible. In particular, $\mathrm{O}(3)$ does not fix any nonzero vector in $\Lambda^4(\mathsf{S}^\ast)$.

Example 2: Here is an example of an $8$-dimensional subspace $N\subset\Lambda^2(\mathbb{R}^8)$ that has no corresponding $a\in\Lambda^4(\mathbb{R}^8)$.

Recall that the compact Lie group $\mathrm{SU}(3)$ has dimension $8$ and its de Rham cohomology is nontrivial only in degrees $0$, $3$, $5$, and $8$. Identify $\mathbb{R}^8$ with ${\frak{g}} = {\frak{su}}(3)$, the tangent space of $\mathrm{SU}(3)$ at the identity matrix, and note that, if $\phi\in\Omega^3\bigl(\mathrm{SU}(3)\bigr)$ is a bi-invariant $3$-form representing a nonzero element of $H^3_{dR}\bigl(\mathrm{SU}(3)\bigr)$, then the value of $\phi$ at the identity is the Cartan $3$-form $$ \kappa(x,y,z) = -\beta\bigl(x,[y,z]\bigr), $$ where $\beta:{\frak{g}}\times{\frak{g}}\to\mathbb{R}$ is the Killing form (which is nondegenerate and allows us to identify $\frak{g}$ with $\frak{g}^\ast$ in a natural way). Now set $$ N = \{i_x\kappa\ |\ x\in \frak{g} \}\subset \Lambda^2(\frak{g}^\ast)\simeq\Lambda^2(\mathbb{R}^8), $$ where $i_x$ denotes interior product with $x\in\frak{g}$. Then $N$ has dimension $8$, and it is easy to see, from the multiplication properties of the Lie algebra ${\frak{g}} = {\frak{su}}(3)$, that $\omega^2\not=0$ for any nonzero $\omega\in N$.