  Let $ K$ be a field admitting a Galois extension $L$ of degree $n$, denoting
the Galois group as $G = \gal(L/K)$. Our focus lies on the space $\sym_K(L)$ of
symmetric $K$-bilinear forms on $L$. We establish a decomposition of
$\sym_K(L)$ into direct sum of $K$-subspaces $A^{\sigma_i}$, where $\sigma_i
\in G$. Notably, these subspaces $ A^{\sigma_i}$ exhibit nice constant rank
properties. The central contribution of this paper is a decomposition theorem
for $\sym_K(L)$, revealing a direct sum of $\frac{(n+1)}{2}$ constant rank
$n$-subspaces, each having dimension of $n$. This holds particularly when $G$
is cyclic, represented as $G = \gal(L/K) = \langle\sigma\rangle$. For cyclic
extensions of even degree $n = 2m$, we present slightly less precise but
analogous results. In this scenario, we enhance and enrich these constant
results and show that, the component $ A^{\sigma}$ often decomposes directly
into a constant rank subspaces. Remarkably, this decomposition is universally
valid when $-1 \notin L^{2}$. Consequently, we derive a decomposition of
$\sym_K(L)$ into subspaces of constant rank under several situations. Moreover,
leveraging these decompositions, we investigate the maximum dimension of an
$n$-subspace inside $M(n,K)$ and $S(n,K)$ for various field $K$ where $M(n,K)$
and $ S(n,K)$ denote the vector spaces $(n \times n)$ matrices and symmetric
matrices over $K$, respectively.
