@@ -39,6 +39,20 @@
 FAST_CLASSES = {}
 
 
+def _check_multidim_table(table, max_ndim=1):
+    """Check that ``table`` has only columns with ndim <= ``max_ndim``
+
+    Currently ECSV is the only built-in format that supports output of arbitrary
+    N-d columns, but HTML supports 2-d.
+    """
+    # Check for N-d columns
+    nd_names = [col.info.name for col in table.itercols() if len(col.shape) > max_ndim]
+    if nd_names:
+        raise ValueError(f'column(s) {",".join(nd_names)} with dimension > {max_ndim} '
+                         "cannot be be written with this format, try using 'ecsv' "
+                         "(Enhanced CSV) format")
+
+
 class CsvWriter:
     """
     Internal class to replace the csv writer ``writerow`` and ``writerows``
@@ -1221,6 +1235,20 @@ def __init__(self):
         self.meta = OrderedDict(table=OrderedDict(),
                                 cols=OrderedDict())
 
+    def _check_multidim_table(self, table, max_ndim=1):
+        """Check that ``table`` has only 1-d columns.
+
+        If a reader class supports N-d columns then override this method.
+
+        Parameters
+        ----------
+        table : `~astropy.table.Table`
+            Input table.
+        max_ndim : int
+            Max allowed number of dimensions (default=1)
+        """
+        _check_multidim_table(table, max_ndim)
+
     def read(self, table):
         """Read the ``table`` and return the results in a format determined by
         the ``outputter`` attribute.
@@ -1409,6 +1437,10 @@ def write(self, table):
         # only used by ECSV and is otherwise just a pass-through.
         table = self.update_table_data(table)
 
+        # Check that table has only 1-d columns. Classes that support output of
+        # N-d columns can override this method.
+        self._check_multidim_table(table)
+
         # Now use altered columns
         new_cols = list(table.columns.values())
         # link information about the columns to the writer object (i.e. self)
 
@@ -54,14 +54,6 @@ def write(self, lines):
         if self.splitter.delimiter not in DELIMITERS:
             raise ValueError('only space and comma are allowed for delimiter in ECSV format')
 
-        for col in self.cols:
-            if len(getattr(col, 'shape', ())) > 1:
-                raise ValueError(
-                    f"ECSV format does not support multidimensional column '{col.info.name}'\n"
-                    'One can filter out such columns using:\n'
-                    'names = [name for name in tbl.colnames if len(tbl[name].shape) <= 1]\n'
-                    'tbl[names].write(...)')
-
         # Now assemble the header dict that will be serialized by the YAML dumper
         header = {'cols': self.cols, 'schema': 'astropy-2.0'}
 
 
@@ -169,6 +169,8 @@ def write(self, table, output):
     def _write(self, table, output, default_kwargs,
                header_output=True, output_types=False):
 
+        core._check_multidim_table(table)
+
         write_kwargs = {'delimiter': self.delimiter,
                         'quotechar': self.quotechar,
                         'strip_whitespace': self.strip_whitespace_fields,
 
@@ -344,6 +344,9 @@ def write(self, table):
         """
         Return data in ``table`` converted to HTML as a list of strings.
         """
+        # Check that table has only 1-d or 2-d columns. Above that fails.
+        self._check_multidim_table(table, max_ndim=2)
+
         cols = list(table.columns.values())
 
         self.data.header.cols = cols
 
@@ -473,6 +473,9 @@ def write(self, table):
 
         core._apply_include_exclude_names(table, self.names, self.include_names, self.exclude_names)
 
+        # Check that table has only 1-d columns.
+        self._check_multidim_table(table)
+
         # Now use altered columns
         new_cols = list(table.columns.values())
         # link information about the columns to the writer object (i.e. self)