Describe $\Lambda$ with linear equations in the standard basis of $L(\mathbb{R}^2,\mathbb{R}^3)$ Let $V \subset \mathbb{R}^3$ be the vector space defined by $x_1+x_2+x_3=0$. In the space of linear transformations $L(\mathbb{R}^2,\mathbb{R}^3)$ there's a subspace
$$\Lambda = \{ \phi \in  L(\mathbb{R}^2,\mathbb{R}^3): \text{im}(\phi) \subseteq V \} $$
We want to describe $\Lambda$ with linear equations in the standard basis of $L(\mathbb{R}^2,\mathbb{R}^3)$.
First of all, $V$ is spanned by $(-1,1,0)$ and $(-1,0,1)$. $\Lambda$ can be described as a subspace of 3 by 2 matrices. Since $\text{im}(\phi) \subseteq V$ these matrices will have linear combinations of $(-1,1,0)$ and $(-1,0,1)$ in their columns. The matrices in the standard basis of $L(\mathbb{R}^2,\mathbb{R}^3)$ would be
$$
        \begin{bmatrix}
        1 & 0 \\
        0 & 0 \\
        0 & 0 \\
        \end{bmatrix},
        \begin{bmatrix}
        0 & 1 \\
        0 & 0 \\
        0 & 0 \\
        \end{bmatrix},
        \begin{bmatrix}
        0 & 0 \\
        1 & 0 \\
        0 & 0 \\
        \end{bmatrix},
        \begin{bmatrix}
        0 & 0 \\
        0 & 1 \\
        0 & 0 \\
        \end{bmatrix},
        \begin{bmatrix}
        0 & 0 \\
        0 & 0 \\
        1 & 0 \\
        \end{bmatrix},
        \begin{bmatrix}
        0 & 0 \\
        0 & 0 \\
        0 & 1 \\
        \end{bmatrix}
$$
Now, my suspicion is that elements of $\Lambda$ can be written down as
$$
(-c_1-c_2)
        \begin{bmatrix}
        1 & 0 \\
        0 & 0 \\
        0 & 0 \\
        \end{bmatrix}
+(-c_3-c_4)
        \begin{bmatrix}
        0 & 1 \\
        0 & 0 \\
        0 & 0 \\
        \end{bmatrix}
+c_1
        \begin{bmatrix}
        0 & 0 \\
        1 & 0 \\
        0 & 0 \\
        \end{bmatrix}
+c_4
        \begin{bmatrix}
        0 & 0 \\
        0 & 1 \\
        0 & 0 \\
        \end{bmatrix}
+c_2
        \begin{bmatrix}
        0 & 0 \\
        0 & 0 \\
        1 & 0 \\
        \end{bmatrix}
+c_3
        \begin{bmatrix}
        0 & 0 \\
        0 & 0 \\
        0 & 1 \\
        \end{bmatrix}
$$
How can I get from this to a system of linear equations?

I'm not sure what exactly is meant with "describe $\Lambda$ in linear equations" (see comments to the question), here's an idea...
A linear transformation $\phi$ in $L$ has a matrix of the form:
$$\begin{bmatrix}
a & b \\
c & d \\
e & f
\end{bmatrix}$$
An element $\mathbf{w} \in \text{im}(\phi)$ is of the form:
$$\mathbf{w}=\begin{bmatrix}
a & b \\
c & d \\
e & f
\end{bmatrix}\begin{bmatrix}
x  \\
y 
\end{bmatrix}=\begin{bmatrix}
ax+by  \\
cx+dy  \\
ex+fy
\end{bmatrix}$$
Now $\phi \in \Lambda$ if $\text{im}(\phi) \subseteq V$ so you need that $\mathbf{w} \in \text{im}(\phi) \implies \mathbf{w} \in V$. But $\mathbf{w} \in V$ means that for all $x$ and $y$; you need:
$$\begin{align}
\underbrace{(ax+by)}_{x_1}+\underbrace{(cx+dy)}_{x_2}+\underbrace{(ex+fy)}_{x_3} = 0  & \iff (a+c+e)x+(b+d+f)y = 0 \\
{} & \iff \left\{\begin{array}{l}
\color{blue}{a+c+e=0} \\
\color{blue}{b+d+f=0}
\end{array}\right.
\end{align}$$
