I am currently reading Goerss and Jardine's book 'Simplicial Homotopy Theory and I am struggling to understand some of the definitions the book introduces early on. In particular I am struggling to understand the definition of the $nth$ skeleton $sk_nX$ of a simplicial set $X$ (defined on page 8 as: the subcomplex of $X$ which is generated by the simplices of $X$ of degree $\leq n$), and the $kth$ horn $\Lambda^n_k$ of the simplicial set $\Delta^n$ (defined on page 6 as the subcomplex of $\Delta^n$ which is generated by all faces $d_j(1_n)$ except the $kth$ face $d_k(1_n)$). To aid in developing an understanding of these definitions I am trying to construct simple examples. So lets consider the simplicial set $\Delta^3$ and its $2nd$ skeleton $sk_2\Delta^3$. Then we have that \begin{align} \left(sk_2\Delta^3\right)_{0} & = \text{hom}_{\Delta}(\mathbf{0},\mathbf{3}), \\ \left(sk_2\Delta^3\right)_{1} & = \text{hom}_{\Delta}(\mathbf{1},\mathbf{3}), \text{and}\\ \left(sk_2\Delta^3\right)_{2} & = \text{hom}_{\Delta}(\mathbf{2},\mathbf{3}). \\ \end{align} Working out what the set $\left(sk_2\Delta^3\right)_{3}$ precisely contains is where I am encountering an issue. The best I can possibly seem to guess is that it contains all the maps $f:\mathbf{3} \to \mathbf{3}$ such that $f = d^i\circ s^j$ for some $i$ and $j$. Then I assume $\left(sk_2\Delta^3\right)_{4}$ contains maps $g:\mathbf{4} \to \mathbf{3}$ such that $g = d^i\circ s^j \circ s^k$ for some $i,j,k$. And so on. Is this correct? If not where am I going wrong?
In the case of simplicial horns I am trying to consider the $0th$ horn of $\Delta^3$. Then we have \begin{align} \left(\Lambda_0^3\right)_{0} & = \text{hom}_{\Delta}(\mathbf{0},\mathbf{3}), \\ \left(\Lambda_0^3\right)_{1} & = \text{hom}_{\Delta}(\mathbf{1},\mathbf{3}), \text{and} \\ \left(\Lambda_0^3\right)_{2} & = \text{hom}_{\Delta}(\mathbf{2},\mathbf{3})\setminus\{d^0\}. \\ \end{align} Working out what maps the set $\left(\Lambda_0^3\right)_{3}$ precisely contains is where I am struggling. Using my above idea figure that $\left(\Lambda_0^3\right)_{3}$ contains all maps $f:\mathbf{3} \to \mathbf{3}$ such that $f = d^i\circ s^j$ for some $i \not = 0$ and $j$. Then I assume $\left(\Lambda_0^3\right)_{4} $ contains maps $g:\mathbf{4} \to \mathbf{3}$ such that $g = d^i\circ s^j \circ s^k$ for some $i \not = 0,j,k$. And so on. Again, is this correct? If not where am I going wrong?
I fear I have totally misunderstood the definitions and I am struggling with finding a source that gives me a simple example of both that I can understand. Any help fixing my above examples and improving my understanding of the definitions overall would be greatly appreciated. Thanks!