Multiple split equations

You need \def not \let \nd you need to get into math mode:

\documentclass{article}
\usepackage[fleqn]{amsmath}

\newenvironment{multeq}{\[%
    \def\n{\\}%
    \def\\{\end{aligned}\\\begin{aligned}}%
    \begin{aligned}%
}{%
    \end{aligned}%
    \]}

\begin{document}

    \begin{gather}
        \begin{aligned}
            S(A) &= (I_{n-k, k}AI_{n-k,k})_{i,j} = (I_{n-k,k})_{i,i}A_{i,j}(I_{n-k,k})_{j,j}\\
            &= \begin{cases}A_{i,j}&\begin{aligned}&(i\leq n-k\wedge j\leq n-k)\\\vee&(i>n-k\wedge j>n-k)\end{aligned}\\-A_{i,j}&\text{otherwise}\end{cases}
        \end{aligned}\\
        \Rightarrow H = S(U(n-k)\times U(k))
    \end{gather}

    \begin{multeq}
        S(A) &= (I_{n-k, k}AI_{n-k,k})_{i,j} = (I_{n-k,k})_{i,i}A_{i,j}(I_{n-k,k})_{j,j}\n
        &= \begin{cases}A_{i,j}&\begin{aligned}&(i\leq n-k\wedge j\leq n-k)\n\vee&(i>n-k\wedge j>n-k)\end{aligned}\n-A_{i,j}&\text{otherwise}\end{cases}\\
    \Rightarrow H = S(U(n-k)\times U(k))
    \end{multeq}

\end{document}

You may need some other changes, depending quite wh\t input markup you want, the above produces